một số lớp hệ phương trình cặp và ứng dụng bản tóm tắt tiếng anh

MINISTRY OF EDUCATION AND TRAINING THAI NGUYEN UNIVERSITY NGUYEN THI NGAN SOME CLASSES OF DUAL EQUATIONS SYSTEMS AND APPLICATIONS Specialization: Mathematies Analysis Code: 62 46 01 02 Scientific supervisors : Dr Nguyen Van Ngoc Associate Prof Dr Ha Tien Ngoan SUMMARIED PHD THESIS OF EDUCATOLOGY Thai Nguyen, 2013 i Introduction Dual equations and systems of dual equations appear when one solves mixed boundary value problems of mathematical physics by using suitable integral transforms Many problems in elasticity theory, problems of crack theory, problems on contact, may reduced to solving different dual equations However, up to now, qualitative resuls on dual equations remain yet restricted and solvability of these equations has been a litle investigated Investigation of systems of dual equations will extend their applications in studing mixed boundary value problems of mathematical physics Therefore, this field of research is well needed Amog analytical methods of resolving mixed boundary value problems the method of dual equations is more general and more vivacious The pioneer works of this methed are ones of Beltrami E, Boussinesq J and Abramov V M The methed then has been developed by Tranter C., Cooke J., Sneddon I., Ufliand Ia S., Babloian A A., Valov G N., Mandal B N., Aleksandrov B M., The general dual equations can be formulated as follows: Let J be finite or infinite interval of R and T be some integral transform on J together with its inverse one T −1 Denote by vb(ξ) the image function v(x) via T The integral dual equations associated to the transform T is of following form ( pT −1 [A(ξ)b u(ξ)](x) = f (x), x ∈ Ω, (1) p0 u(x) = g(x), x ∈ Ω0 , u b(ξ) = A2 (ξ)b v (ξ), where Ω, Ω0 are systems of distinct intervals of J such that Ω ∪ Ω0 = J, p A1 (ξ) and p0 are recpectively restriction operators on Ω and Ω0 , A(ξ) = , A2 (ξ) A1 (ξ) and A2 (ξ) are both known The function A(ξ) is called the symbol of pseudo-differential operator: Au := T −1 [A(ξ)b u(ξ)](x) We note that the dual equations (1) is indeed the Dirichlet problem in Ω for the pseudodifferential equation Au = f (x) in Ω mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh During the last 50 year there appeared different metheds for resolving many dual integral equations which are associated to various integral transfoms In general, these metheds seemed to look formal Thats means these metheds did not investigate solvability of dual equations and did not have strict garantees for mathematical transformations However, these matheds had stimulated developenment of the theory of dual equations assoaated to diffrent integral transforms In the case where (1) T is the Fourier transp form and A(ξ) = ξ − k , Ω = (−1, 1), g(x) = 0, Eswaran (1990) has propased a method of solving for corrsponding dual equations Sneddon I S (1966) and several other authors had investigated dual equations where kernels of T contain trigonometric functions Different methods, which are used in solving the dual integral equations, can be classified into following groups of approach - Approach from analysis and functional analysis - Approach from the theory of distributions - Approach from the theory of pseudodifferential operators acting on Sobolev spaces In 1975, Walton J R using approach by Zemanian’s distributions considered uniqueness of solutions to Titchmarsh’s dual equations Then, this approach had been used by other authors In 2000 Ciaurri O’., Gaidalupe Jose’ J., Pere’z Mario and Varona Juan L considered solvability of these equations in Lebesgue spaces In 1986, Nguyen Van Ngoc and Popov G Ya innestigated solvability of dual integral equations with trigonometric kernels on a system of intervals In 1988, using the approach by pseudo-differential operators Nguyen Van Ngoc studied solvability of dual Fourier integral equations in approppriate Sobolev spaces In 2009, Nguyen Van Ngoc had exhibited a methed of solution to following dual Fourier integral equations with symbols of increasing degree p∈N: ( F −1 [|ξ|p A(ξ)b u(ξ)](x) = f (x), x ∈ (a, b), (2) −1 F [b u(ξ)](x) = 0, x ∈ R \ (a, b) where u b(ξ) ∈ S ∩ C ∞ (R) being unkown, f (x) in known from Sobolev space H −p/2 (a, b), A(ξ) is known Recently, in the frame of this Thesis, Nguyen Van Ngoc and Nguyen Thi mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh Ngan have obtained some results on solvability to systems of dual Fourier integral equations ( pF −1 [A(ξ)b u(ξ)](x) = f (x), x ∈ Ω, (3) p0 F −1 [b u(ξ)](x) = g(x), x ∈ Ω0 = R \ Ω, b is vector-function to be found, f , g where Ω is a finite interval of R, u are known vector-functions, A(ξ) is a square definite matrix, that is called symbol of the system (3), p and p0 are respectively restriction operator on Ω and Ω0 The Thesis consists of Introduction, chapters, Conclusion and Cited literature The Chapter is devoted to exhibit general theory of system of n dual Fourier integral equations on a system of distinct intervals of the real axis The Paragraphs 1.1-1.3 present some basis notions: Fourier transform of fast decreasing functions and tempered distributions, Sobolev spaces of arbitrary read order s The Paragraphs 1.4 presents vector Sobolev H~s (R), H~s◦ (Ω), H~s◦,◦ (Ω), H~s (Ω) The isomorphism between H−~s (R) and (H~s (R))∗ is established, which is dual to H~s (R) ( Theorem 1.8) Analoguesly, the general from of linear continuous functional on H~◦s (Ω) is described (Theorem 1.9) The Paragraphs 1.5 is devoted to study pseudo-differential operators of from (Au)(x) := F −1 [A(ξ)b u(ξ)](x) The matrix A(ξ) is called symbol of operator A Based on many concret dual equations, we have introduced ~ some classes of symbols such as: Σα~ (R), Σα+ (R), Σα◦~ (R) ( Definition 1.11) In 1.6 we study solvability of the dual equations system (3) under following hypothesis ( A(ξ) ∈ Σα◦~ (R), A(ξ) is positively definite ξ ∈ R, (4) f (x) ∈ H−~α/2 (Ω), g(x) ∈ Hα~ /2 (Ω0 ), b (ξ) is found in form u b (ξ) = F [u](ξ), where u ∈ Hα~ /2 (R) the unknown u Chapter is devoted to present some classes of dual integral equations, to which several of mixed boundary value problem for harmonic or biharmonic in a strip have been reduced The Paragraph 2.1 considers a dual integral equations systems with symbol of increasing degree 1, that appears in studying first mixed boundary mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh value problem for harmonic equation in a strip The existence and uniqueness of solutions of these dual equations in Sobolev spaces have been proved (Theorem 2.1 and Theorem 2.2) ( The Subsection 2.1.3) In the Subsection 2.1.4 we represent solutions um (x), m = 1, via auxilary appopriate ones Z b vm (t)sign(x − t)dt, vm ∈ L2ρ (a, b) ⊂ H◦−1/2 (a, b), m = 1, 2, um (x) = a by this transform, the dual equations system is reduced to singular integral equations system with Cauchy kernels By using orthogonal polynomial method the last system of equations have been reduced to a infinite system of linear equations The Theorem 2.6 says that the infinite system of linear equations has a unique solution in the space `2 (Subection 2.1.5) The Paragraph 2.2 deals with a dual integral equations system, that appears in studying a mixed boudary value problem for a strip of elastcity By using representation of displacement via two harmonic functions, the problem of elasticity has been reduced to a dual equations system with symbol of increasing degree as in 2.1, but its structure is more complicated The Paragraph 2.3 studies a dual equations system with symbol of decreasing degree 1, that appears in solving mixed boundary value problem for equation in a strip The Paragraph 2.4 is devoted to study a dual equations systems with symbol of increasing-decreasing degree 1, that appears in solving second mixed boundary value problem for harmonic equationin a strip In Chapter we solve approximately the dual integral equations system, that has been investigated in the Paragraph 2.1 mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh Chapter General dual Fourier integral equations systems This chapter presents general theory for systems of n- dual Fourier integral equations These systems of dual equations have been generalised and abstracted from many concret ones, that appear in solving various mixed boundary value problems of mathematical physics Some concret dual Fourier integral equations systems will be investigated in Chapter 1.1 Fourier transform for fast decreasing test functions This paragraph presents some basic notions and reluts on Fourier transform for functions from S 1.2 Fourier transform for tempered distributions This paragraph presents some basic notions and relusts on Fourier transform for distributions from S 1.3 Sobolev spaces This paragraph present some basic notions and relusets on Sobolev s spaces of arbitrary real order s ∈ R as H s (R), H◦s (Ω), H◦,◦ (Ω), H s (Ω) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 1.4 Sobolev vectors spaces 1.4.1 Notion Let X be a linear topological space We denote the direct product of n elements X by X n A topology in X n is given by the usual topology of the direct product We shall use bold letters for denoting vector-values and matrices Denote by u a vector of the form u = (u1 , u2 , , un ), and Sn = S × S × S, S n = S × S × S s s For the vectors u, v ∈ (S0 )n Let H sj (R), H◦ j (Ω), H◦,◦j (Ω), H sj (Ω) be the Sobolev spaces, where j = 1, 2, , n; Ω is a certain set of intervals in R We put ~s = (s1 , s2 , , sn )T , H~s (R) = H s1 (R) × H s2 (R) × × H sn (R), H~◦s (Ω) = H◦s1 (Ω) × H◦s2 (Ω) × × H◦sn (Ω), s s1 s1 sn H~◦,◦ (Ω) = H◦,◦ (Ω) × H◦,◦ (Ω) × × H◦,◦ (Ω), H~s (Ω) = H s1 (Ω) × H s1 (Ω) × × H sn (Ω) The scalar product and a norm in H~s and H~◦s (Ω) are given by the formulas (u, v)~s = n X (uj , vj )sj , ||u||~s = n X j=1 ||uj ||2sj 1/2 , (1.1) j=1 s where (uj , vj )sj and ||uj ||sj are given by the formulas in H sj , H◦ j (Ω) The norm in H~s (Ω) is defined by the equation ||u||H~s (Ω) := n X ||uj ||2H sj (Ω) 1/2 (1.2) j=1 1.4.2 The linear continuous functional Theorem 1.8 Let (H~s (R))∗ be a dual space of the space H~s (R) Then (H~s (R))∗ is isomorphic to the space H−~s (R) Moreover, a value of a functional f ∈ H−~s (R) on an element u ∈ H ~s (R) is given by the formula n Z ∞ X uj (ξ)dξ, (1.3) (f , u)◦ = fbj (ξ)b j=1 −∞ mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh where u bj (ξ) = F [uj ](ξ), fbj (ξ) = F [fj ](ξ) Theorem 1.9 Let Ω ⊂ R, u = (u1 , u2 , , un )T ∈ H~s (Ω), f ∈ H−~s (Ω) and `f = (`1 f1 , `2 f2 , , `n fn )T be an extension of f from Ω to R belonging H−~s (R) Then the integral n Z ∞ X uj (ξ)dξ (1.4) `d [f , u] := (`f , u)◦ := j fj (ξ)b j=1 −∞ does not depend on the choice of the extension `f Therefore, this formula defines a linear continuous functional on H~s◦ (Ω) Conversely, for every linear continuous functional Φ(u) on H~◦s (Ω) there exists an element f ∈ H−~s (Ω) such that Φ(u) = [f , u] and ||Φ|| = ||f ||H−~s (Ω) 1.5 Pseudo-differential operators 1.5.1 Notion b (ξ) = F [u](ξ) The Definition 1.10 Let A(ξ) ∈ Σα~ (R), u ∈ H~s , u operators A is given by the formula: (Au)(x) := F −1 [A(ξ)b u(ξ)](x), x ∈ Rn , (1.5) where A(ξ) = ||aij (ξ)||n×n is a square matrix of order n, u = (u1 , u2 , , un )T b (ξ) := is a vector, transposed to the line vector (u1 , u2 , , un ), and u T F [u](ξ) = (F [u1 ], F [u2 ], , F [un ]) and the matrixA(ξ) is symbol of pseudodifferential operators A Definition 1.11 Let A(ξ) = ||aij (ξ)||n×n , ξ ∈ R be a square matrix order n, where aij (ξ) are continuous functions on R, αj ∈ R, (i = 1, 2, , n), α ~ = (α1 , α2 , , αn )T Denote by A(ξ) = ||aij (ξ)||n×n , the class of square matrices Σα~ (R) such that aii (ξ) ∈ σ αi (R), aij (ξ) ∈ σ βij (R), βij ≤ (αi + αj ), (1.6) ~ we shall say that the matrix A(ξ) belongs to the class Σα+ (R), if A(ξ) ∈ α ~ T Σ (R) and it is Hemitian, i.e (A(ξ)) = A(ξ), and satisfies the condition: T w Aw ≥ C1 n X (1 + |ξ|)αj |wj |2 , w = (w1 , w2 , , wn )T ∈ Cn , j=1 mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh (1.7) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh where C1 is a positive constant and w is the complex conjugate of w Finally, we say that the matrix A(ξ) ∈ Σα~ (R) belongs to the class Σα◦~ (R), if RewT Aw ≥ 0, w = (w1 , w2 , , wn )T ∈ Cn , (1.8) moreover, RewT Aw = it is positive-definite for almost everywhere 1.5.2 The scalar product and a norm ~ Proposition 1.5 Let the matrix A(ξ) = A+ (ξ) belong to the class Σα+ (R) α ~ /2 Then the scalar product and norm in H (R) may be defined by the formulas Z ∞ (u, v)A+ ,~α/2 = F [vT ](ξ)A+ (ξ)F [u](ξ)dξ, (1.9) ||u||A+ ,~α/2 = Z −∞ ∞ F [uT ](ξ)A+ (ξ)F [u](ξ)dξ 1/2 , (1.10) −∞ respectively 1.5.3 The imbedding compact Proposition 1.7 Let Ω be a bounded subset of intervals in R Then the imbedding H~s (Ω) into H~s−~ε(Ω) is compact, where ~ε = (ε1 , ε2 , , εn )T > 0(⇔ εj > 0, j = 1, 2, , n) 1.6 1.6.1 Solvability of the systems of dual equations Uniqueness theorem In this section we shall investigate a system of dual equations in the form ( pF −1 [A(ξ)b u(ξ)](x) = f (x), x ∈ Ω, (1.11) p0 F −1 [b u(ξ)](x) = g(x), x ∈ Ω0 := R \ Ω, b (ξ) = (b where Ω is a certain interval in R, u u1 (ξ), u b2 (ξ), , u bn (ξ))T is a function vector to be found, f (x) = (f1 (x), f2 (x), , fn (x))T ∈ (D0 (Ω))n , g(x) = (g1 (x), g2 (x), , gn (x))T ∈ (D0 (Ω0 ))n are given vectors of distributions on Ω and Ω0 respectively, A(ξ) = ||aij (ξ)||n×n is a given square mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh matrix order n and is called the symbol of the system (1.11); p and p0 are restriction operators to Ω and Ω0 respectively We shall consider the system of dual integral equations (1.11) under the following conditions ( A(ξ) ∈ Σα◦~ (R), and A(ξ) is positive-definite for almost ξ ∈ R, f (x) ∈ H−~α/2 (Ω), g(x) ∈ Hα~ /2 (Ω0 ), (1.12) b (ξ) in the form u b (ξ) = F [u](ξ), where u ∈ and shall find the vector u α ~ /2 H (R) Theorem 1.10 (Uniqueness) Under the assumptions (1.12), the system of dual equations (1.11) has at most one solution u ∈ Hα~ /2 (R) 1.6.2 Existence theorem Lemma 1.1 The system of dual integral equations (1.11) is equivalent to the following system of dual equations g(ξ)](x), x ∈ Ω, pF −1 [A(ξ)b v(ξ)](x) = f (x) − pF −1 [A(ξ)`c (1.13) α ~ /2 where v = F −1 [ˆ v] ∈ H◦ (Ω) satisfies the relation v + `0 g = u ∈ Hα~ /2 (R) (1.14) (`0 g being an arbitrary extension of the generalized g from Ω0 into R) g(ξ)](x), we rewrite (1.13) in the form Denote h(x) = f (x) − pF −1 [A(ξ)`c (Av)(x) = h(x), x ∈ Ω (1.15) α ~ The case A(ξ) ∈ Σ+ (R) ~ Theorem 1.11 (Existence) Let h ∈ H−~α/2 (Ω), A(ξ) = A+ (ξ) ∈ Σα+ (R) α ~ /2 Then, the system (1.15) has a unique solution v ∈ H◦ (Ω) We have ||u||A,~α/2 ≤ C(||f ||H−~α/2 (Ω) + ||g||Hα~ /2 (Ω0 ) ), (1.16) where C is a positive α ~ /2 Theorem 1.12 (Existence) Let A(ξ) ∈ Σ+ (R), f ∈ H−~α/2 (Ω), g ∈ H−~α/2 (Ω0 ) Then, the system of dual integral equations (1.11) has a unique solution u = F −1 [ˆ u] ∈ Hα~ /2 (R) and satisfies the estimation (1.16) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 10 The case A(ξ) ∈ Σ◦α~ (R) We assume in addition that the set Ω is bounded in R and there exists ~ a square A+ (ξ) ∈ Σα+ (R) sem order with matrice A(ξ), such that ~ B(ξ) := A(ξ) − A+ (ξ) ∈ Σα~ −β (R), (1.17) where β~ = (β1 , β2 , , βn )T ∈ Rn , βj > (j = 1, 2, , n) Theorem 1.13 (Existence) Let Ω is bound in R Under conditions (1.12) and (1.17) for every f ∈ H−~α/2 (Ω), g ∈ Hα~ /2 (Ω0 ) the system of dual equation (1.11) has a unique solution u = F −1 [ˆ u] ∈ Hα~ /2 (R) • Conclusions of the Chapter The main relutsts of this chapter consist of : - Construc some vector Sobolev H~s (R), H~s◦ (Ω), H~s◦,◦ (Ω), H~s (Ω) The isomorphism between H−~s (R) and (H~s (R))∗ is establised, which is dual to H~s (R) (Theorem 1.8) The general form of linear continuous functional on H~◦s (Ω) is described (Theorem 1.9) - Introduce the definition of vector pseudo-differential operators (Au)(x) := F −1 [A(ξ)b u(ξ)](x) the definitions of some classes of symbols: α ~ α ~ α ~ Σ (R), Σ+ (R), Σ◦ (R) - Prove existence and uniqueness of solutions to dual Fourier integral equations systems in appropriate Sobolev spaces mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh Chapter Dual equations systems of some mixed boundary value problems for harmonic and biharmonic equations in a strip In this chapter we consider some mixed boundary value problems for harmonic and biharmonic equations in a strip By using the Fourier transform with respect to the variable x, these mixed boundary value problems have been reduced to corresponding Foutier integral equations systems This chapter is written on base of the results of our paper, that have been published in Journal of science and technology, Thai Nguyen university, Vietnam Journal of Mathematics, Acta Mathematica Vietnamica and in Proceedings of ”Based on the selected lectures of the 17th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications” 2.1 The first mixed boundary value problem for Laplace equation In this paragraph we will investigate a dual Fourier integral equations systems, with symbol of increasing degree 1, which appears in studying the first mixed boundary value problem for Laplace equation in a strip 2.1.1 Formulation of the problem Consider the following problem: find a solution of the Laplace equation ∂ 2Φ ∂ 2Φ + = 0, ∂x2 ∂y (−∞ < x < ∞, < y < h) 11 mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh (2.1) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 12 subject to the boundary conditions   − ∂Φ(x, 0) = f (x), x ∈ (a, b),  ∂Φ(x, h) = f (x), x ∈ (a, b), ∂y ∂y ,   Φ(x, 0) = 0, x ∈ R \ (a, b), Φ(x, h) = 0, x ∈ R \ (a, b), (2.2) where f1 , f2 are given functions 2.1.2 Reduction to a system of integral equations We shall solve the formulated problem (2.1)-(2.2) by the method of Fourier transforms We have the system of dual integral equations with respect to u b1 (ξ), u b2 (ξ) : ( F −1 [A(ξ)b u(ξ)](x) = f (x), x ∈ (a, b), (2.3) F −1 [b u(ξ)](x) = 0, x ∈ R \ (a, b), b (ξ) = F [u](ξ) = (F [u1 (x)], F [u2 (x)])T (ξ), where f (x) = (f1 (x), f2 (x))T , u u1 (x) = Φ(x, 0), u2 (x) = Φ(x, h),   |ξ| |ξ| cosh(|ξ|h) −  sinh(|ξ|h) sinh(|ξ|h)    A(ξ) =  |ξ| cosh(|ξ|h)  |ξ| − sinh(|ξ|h) sinh(|ξ|h) 2.1.3 Solvability of systems of dual equations (2.3) The system (2.3) can be rewritten ( pF −1 [A(ξ)b u(ξ)](x) = f (x), x ∈ (a, b), p0 F −1 [b u(ξ)](x) = 0, x ∈ R \ (a, b), (2.4) where p and p0 denote restriction operators to (a, b) and R \ (a, b), respectively Denote α ~ = (1, 1)T Theorem 2.1 (Uniqueness) Let f ∈ H−~α/2 (a, b) Then the system (2.4) α ~ /2 has at mots one solution u ∈ H◦ (a, b) Theorem 2.2 (Existence) If f ∈ H−~α/2 (a, b) then the system (2.4) has a α ~ /2 unique solution u ∈ H◦ (a, b) 2.1.4 Reduction of the system of dual equations to the system of singular integral equations with Cauchy kernel Theorem 2.4 The system of dual integral equations (2.3) with respect to (b u1 (ξ), u b2 (ξ)) is equivalent to the following system of singular integral mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 13 equations with Cauchy kernel on (a, b) :  Z b Z b Z  b v1 (t)dt   + v1 (t)`11 (x − t)dt + v2 (t)`12 (x − t)dt = −if1 (x),    πi Za x − t a a Z Z b b b v2 (t)dt + v1 (t)`21 (x − t)dt + v2 (t)`22 (x − t)dt = −if2 (x),    πi a x − t a a    v (t) ∈ L2 (a, b), m = 1, 2; a < x < b, m ρ (2.5) with vm ∈ O1 (a, b), i.e Z b vm (x)dx = 0, (m = 1, 2), (2.6) a where um (x) = Z b vm (t)sign(x − t)dt, x ∈ R, (m = 1, 2), (2.7) a Z −i ∞ e−ξh `11 (x) = `22 (x) = sin(ξx)dξ, π sinh(ξh) Z i ∞ sin(ξx) `12 (x) = `21 (x) = dξ π sinh(ξh) 2.1.5 Reduction of the system of singular integral equations with Cauchy kernel to an infinite system of linear algebraic equations φm (t) Put vm (t) = , we expand the functions φm (t) to series φm (t) = ρ(t) ∞ X (m) (m) (m) Aj Tj [η(t)], where Aj are unknown constans, besides, {Aj }∞ j=1 ∈ j=1 `2 (m = 1, 2) Substituting φm (t) into the system of singular integral equations, change order of integraltions and summations, using the orthogonality of Chebuyshev polynomials Uj we obtain ∞ −(b − a) (m) X X (k) (mk) An+1 + Aj Cnj = Fn(m) , 4i j=1 (m = 1, 2), (n = 0, 1, 2, ), k=1 (2.8) where (mk) Cnj Z = b Z ρ(x)Un [η(x)]dx a a b Tj [η(t)] lmk (x − t)dt, ρ(t) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh (2.9) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 14 Fn(m) Z = −i b ρ(x)Un [η(x)]fm (x)dx, (m = 1, 2), (n = 0, 1, 2, ) a (2.10) (1) (2) We introduce notations X2n−1 = An , X2n = An , (n = 1, 2, 3, ), 4i 4i E2n+1 = − Fn(1) , E2n+2 = − Fn(2) , (n = 0, 1, 2, ), b−a b−a 4i 4i (11) (12) Cnj , C2n,2j+1 = − Cnj , C2n−1,2j+1 = − b−a b−a 4i 4i (21) (22) C2n−1,2j+2 = − Cnj , C2n,2j+2 = − Cnj b−a b−a Then the system (2.8) can be written in form ∞ X Cn,j Xj = En (n = 1, 2, ) Xn + (2.11) (2.12) (2.13) (2.14) j=1 Theorem 2.6 Let f1 (x) and f2 (x) be the those functions, such that the set {En }∞ n=1 defined by (2.10), (2.11) belong `2 Then the infinite system (2.14) possesses a unique solution {Xn }∞ n=1 ∈ `2 This infinite system is quasi-completely 2.2 A mixed boundary value problem of elasticity theory for a strip 2.2.1 Formulation of the problem Consider the following problem: find Φ(x, y) and Ψ(x, y) such that ∂ 2Φ ∂ 2Φ ∂ 2Ψ ∂ 2Ψ + = 0, + = 0, ∂x2 ∂y ∂x2 ∂y subject to the boundary conditions (−∞ < x < ∞, < y < h) (2.15) τxy (x, h) = τ0 (x), σy (x, h) = σ0 (x), −∞ < x < ∞, ( τxy (x, 0) = σy (x, 0) = 0, x ∈ (a, b), u(x, 0) = v(x, 0) = 0, x ∈ R \ (a, b), (2.16) (2.17) where τ0 (x), σ0 (x) are given functions, u(x, y), v(x, y) are horizontal and vertical displacements respectively, σy (x, y), τxy (x, y) are tangent and normal pressures respectively, µ and ν are Lame’s constant and poisson coeffient, respectively (µ > 0, < ν < 1/2) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 15 2.2.2 Reduction to a system of integral equations We have the system of dual integral equations with respect to u b1 (ξ), u b2 (ξ) ( F −1 [|ξ|A0 (ξ)b u(ξ)](x) = f (x), x ∈ (a, b), (2.18) −1 F [b u(ξ)](x) = 0, x ∈ Ω0 = R \ (a, b), b (ξ) = F [u(x)](ξ) where u1 (x) = 2µu(x, 0), u2 (x) = 2µv(x, 0), u T = (F [u1 (x)], F [u2 (x)]) (ξ), f (x) = (f1 (x), f2 (x))T   a11 (ξ) isign(ξ).a12 (ξ) , A0 (ξ) =  −isign(ξ).a21 (ξ) a22 (ξ) with 2(1 − ν)[cosh(|ξ|h) sinh(|ξ|h) + |ξ|h] , 4(1 − ν)2 + |ξ|2 h2 + (3 − 4ν) sinh2 (|ξ|h) (1 − 2ν) sinh2 (|ξ|h) + |ξ|2 h2 a21 (ξ) = a12 (ξ) = , 4(1 − ν)2 + |ξ|2 h2 + (3 − 4ν) sinh2 (|ξ|h) 2(1 − ν)[cosh(|ξ|h) sinh(|ξ|h) − |ξ|h] a22 (ξ) = 4(1 − ν)2 + |ξ|2 h2 + (3 − 4ν) sinh2 (|ξ|h) a11 (ξ) = 2.2.3 Solvability of systems of dual equations (2.18) The system (2.18) can be rewritten ( pF −1 [|ξ|A0 (ξ)b u(ξ)](x) = f (x), x ∈ (a, b), −1 p F [b u(ξ)](x) = 0, x ∈ R \ (a, b), (2.19) where p and p0 denote restriction operators to (a, b) and R \ (a, b), respectively Theorem 2.7 (Uniqueness) Let f ∈ H−~α/2 (a, b) Then the system (2.19) α ~ /2 has at mots one solution u ∈ H◦ (a, b) Theorem 2.8 (Existence) Let τ0 (x) and σ0 (x) be such that the function f (x) = (f1 (x), f2 (x))T belongs to H−~α/2 (a, b), α ~ = (1, 1)T Then the system α ~ /2 of dual equation (2.19) has a unique solution u = F −1 [b u] ∈ H◦ (a, b), i.e u(x, 0) ∈ H◦1/2 (a, b), v(x, 0) ∈ H◦1/2 (a, b), where u(x, 0) and v(x, 0) are horizontal and vertical displacements on the axis y = 0, respectively mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 16 2.2.4 Reduction of the system of dual equations to the system of singular integral equations with Cauchy kernel This subsection as the Subsection 2.1.4 2.2.5 Reduction of the system of singular integral equations with Cauchy kernel to an infinite system of linear algebraic equations This subsection as the Subsection 2.1.5 2.3 A mixed boundary value problem for harmonic equation In this paragraph we will investigate a dual Fourier integral equations system with symbol of decreasing degree 1, which appear in studying mixed boundary value problem for biharmonic equation in a strip 2.3.1 Formulation of the problem Consider the following problem: find Φ(x, y) of the harmonic equation ∂ 4Φ ∂ 4Φ ∂ 4Φ ∆ Φ(x, y) = +2 2 + =0 ∂x4 ∂x ∂y ∂y subject to the boundary conditions Φy=0 = r1 (x), x ∈ R, Φy=h = r2 (x), x ∈ R,   ∂Φ = f1 (x), x ∈ (−a, a), ∂y y=0 M [Φ] = 0, x ∈ R \ (−a, a), y=0 where (2.20) (2.21)   ∂Φ = f2 (x), x ∈ (−a, a), ∂y y=h , M [Φ] = 0, x ∈ R \ (−a, a), y=h (2.22) ∂ 2Φ ∂ 2Φ M [Φ] = M [Φ](x, y) = + ν , < ν < ∂y ∂x mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh (2.23) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 17 2.3.2 Reduction to a system of integral equations We have the system of dual integral equations with respect to u b1 (ξ), u b2 (ξ) : ( F −1 [A(ξ)b u(ξ)](x) = e f (x), x ∈ (−a, a), (2.24) F −1 [b u(ξ)](x) = 0, x ∈ R \ (−a, a), b (ξ) = F [u(x)](ξ), where u1 (x) = M [Φ](x, 0), u2 (x) = M [Φ](x, h), u T e f (x) = (fe1 (x), fe2 (x)) , fe1 (x) = −f1 (x) − F −1 [a1 (ξ)b r1 (ξ)](x) + F −1 [a2 (ξ)b r2 (ξ)](x), (2.25) fe2 (x) = f2 (x) + F −1 [a2 (ξ)b r1 (ξ)](x) − F −1 [a1 (ξ)b r2 (ξ)](x), (2.26)   sinh(|ξ|h) cosh(|ξ|h) − |ξ|h |ξ|h cosh(|ξ|h) − sinh(|ξ|h)   2|ξ| sinh2 (|ξ|h) 2|ξ| sinh2 (|ξ|h)  A(ξ) =   |ξ|h cosh(|ξ|h) − sinh(|ξ|h) sinh(|ξ|h) cosh(|ξ|h) − |ξ|h  2|ξ| sinh2 (|ξ|h) 2|ξ| sinh2 (|ξ|h) 2.3.3 Solvability of systems of dual equations (2.24) The system (2.24) can be rewritten ( pF −1 [A(ξ)b u(ξ)](x) = e f (x), p0 u := p0 F −1 [b u(ξ)](x) = 0, x ∈ (−a, a), x ∈ R \ (−a, a) (2.27) where p and p0 denote restriction operators to (−a, a) and R \ (−a, a), respectively We have r1 (x) and r2 (x) ∈ H (R), f1 (x) and f2 (x) ∈ H (−a, a), e f (x) ∈ H−~α/2 (−a, a), α ~ = (−1, −1)T (2.28) (2.29) Theorem 2.13 (Uniqueness) Let conditions (2.29) be fulfiled Then the α ~ /2 system of dual equations (2.27) has at most one solution u ∈ H◦ (−a, a) Theorem 2.14 (Existence) Let assumptions (2.28) hold Then the system α ~ /2 of dual equations (2.27) has a unique solution u = F −1 [b u] ∈ H◦ (−a, a) mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 18 2.3.4 Reduction of the system of dual equations to the system of singular integral equations with logarithm kernel We rewritten (2.24) in the form  i h ∗  X   −1 amn (ξ) F u bn (ξ) (x) = fem (x), x ∈ (−a, a), |ξ| n=1   um (x) = F −1 [b um ](x) = 0, x ∈ R \ (−a, a), (m = 1, 2), (2.30) where a∗11 (ξ) = a∗22 (ξ) = |ξ|a11 (ξ), a∗12 (ξ) = a∗21 (ξ) = |ξ|a12 (ξ), fem (x), m = 1, 2, are given by the formulas (2.25) v`a (2.26) Theorem 2.16 The system of dual equations (2.30) with respect to (b u1 (ξ), u b2 (ξ)) is equivalent to the following system of integral equations on (−a, a) :  Z Z a a X    ln um (t)k1m (x − t)dt = fe1 (x), u1 (t)dt +   2π x − y −a m=1 −a Z Z a a X  1   ln um (t)k2m (x − t)dt = fe2 (x), m = 1,  u2 (t)dt +  2π x−y −a m=1 −a Z ∞ a∗12 (ξ) cos(ξx)dξ, where k12 (x) = k21 (x) = π ξ Z ∞ ∗ πx 2a11 (ξ) − tanh(ξh) knn (x) = ln x coth + cos(ξx)dξ, n = 1, 2π 4h 2π ξ 2.3.5 Reduction of the system of singular integral equations with logarithm kernel to an infinite system of linear algebraic equations This subsection as the Subsection 2.1.5 2.4 The second mixed boundary value problem for Laplace equation In this paragraph we will investigate a dual Fourier integral equations systems, with symbol of increasing-decreasing degree 1, which appears in studying the first mixed boundary value problem for Laplace equation in a strip mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 19 2.4.1 Formulation of the problem Consider the following problem: find a solution of the Laplace equation ∂ 2Φ ∂ 2Φ + = 0, ∂x2 ∂y (−∞ < x < ∞, < y < h) (2.31) subject to the boundary conditions    ∂Φ (x, h) = f (x), x ∈ (a, b), −Φ(x, 0) = f1 (x), x ∈ (a, b), ∂y , ∂Φ   (x, 0) = 0, x ∈ R \ (a, b), Φ(x, h) = 0, x ∈ R \ (a, b), ∂y (2.32) where f1 , f2 are given functions 2.4.2 Reduction to a system of dual integral equations We have the system of dual integral equations with respect to u b1 (ξ), u b2 (ξ) : ( F −1 [A(ξ)b u(ξ)](x) = f (x), x ∈ (a, b), (2.33) F −1 [b u(ξ)](x) = 0, x ∈ R \ (a, b), b (ξ) = F [u](ξ) = (F [u1 (x)], F [u2 (x)])T , where f (x) = (f1 (x), f2 (x))T , u   tanh(|ξ|h) −  |ξ| cosh(|ξ|h)    A(ξ) =   |ξ| tanh(|ξ|h) cosh(|ξ|h) 2.4.3 Solvability of systems of dual equations (2.33) The system (2.33) can be rewritten: ( pF −1 [A(ξ)b u(ξ)](x) = f (x), x ∈ (a, b), p0 F −1 [b u(ξ)](x) = 0, x ∈ R \ (a, b), (2.34) where p and p0 denote restriction operators to (a, b) and R \ (a, b), respectively Theorem 2.19 (Uniqueness) Let f ∈ H−~α/2 (a, b) Then the system (2.34) α ~ /2 has at mots one solution u ∈ H◦ (a, b), α ~ = (−1, 1)T Theorem 2.20 (Existence) If f ∈ H−~α/2 (a, b) then the system (2.34) has α ~ /2 a unique solution u ∈ H◦ (a, b), α ~ = (−1, 1)T mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 20 2.4.4 Reduction of the system of dual equations to the system of singular integral equations with Cauchy kernel dv1 (x) 1/2 , v1 ∈ H◦ (a, b) ∩ L2ρ−1 (a, b) and We put u1 (x) = dx Z b u2 (x) = v2 (t)sign(x − t)dt, v2 ∈ L2ρ (a, b) ⊂ H◦−1/2 (a, b) we reduction a system (2.33) to the system of singular integral equations with Cauchy kernel on (a, b) as in 2.1.4 2.4.5 Reduction of the system of singular integral equations with Cauchy kernel to an infinite system of linear algebraic equations This subsection as the Subsection 2.1.5 • Conclusion of the Chapter In this chapter we investigate some classes of dual Fourier integral equations systems that appear in studying mixed boundary value problems for harmonic and biharmonic equations in a strip The main steps in our investigation are following: - Reduce mixed boundary problems to corresponding dual equations systems, prove theorems on existence and uniqueness of their solutions in appropriate Sobolev spaces; - Reduce dual equations systems to systems singular integral equations with Cauchy kernel or logarithm kernel; - Futher reduction of systems of singular integral equations to infinite systems of linear equations Prove that the last systems has unique solution in `2 and they are quasi-completely regular mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh Chapter Approximate solutions to a singular integral equations system of a dual Fourier integral equations system In this chapter we solve approximately a singular intergral equations system of a dual equations sysytem, that arises in studying mixed boundury value problem for the harmonic equation in Paragraph 2.1 3.1 Reducing singular integral equation to dimensionness from We transform singular integral equation systems with Cauchy kernel (2.5) into following dimensionness ones:  Z ∗ Z  X  vm (τ ) ∗ ∗ 1 dτ + Kmk (y − τ )vk∗ (τ )dτ = fm (y), π −1 y − τ (3.1) −1 k=1    −1 < y < 1, m = 1, 2, where λ ∗ ∗ K11 (y − τ ) = K22 (y − τ ) = π Z ∞ e−z sin[zλ(y − τ )]dz, sinh z −λ ∞ − τ) = − τ) = sin[zλ(y − τ )]dz, π sinh z 2x − b − a 2t − b − a b−a y= , τ= , z = ξh, λ = b−a b−a 2h ∗ K12 (y Z ∗ K21 (y 21 mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 22 3.2 Approximate solutions to singular integral equations system 3.2.1 Approximate kernels of singular integral equations system By using Chebyshev-Laguerre formula, we get approximate kernel ma∗ ∗ (y −τ ) for singular integral equation system (2.5) (y −τ ) and K12 trieces K11 3.2.2 Approximate solutions to singular integral equations system In this paragraph we solve approximately dimensionness singular integral ∗ ∗ equations system (3.1) where the kernels K11 (y − τ ) and K12 (y − τ ) are calculated with n = 4, and the right-hand sides are polynomial of degree More prisely, we consider following problem Problem Look for solutions to dimensionness singular integral equations system (3.1) satisfying conditions Z Z ∗ v1 (τ )dτ = 0, v2∗ (τ )dτ = (3.2) −1 −1 ∗ ∗ The kernels K11 (y − τ ) and K12 (y − τ ) are calculated with n = 4, f1∗ (y) = a0 + a1 y + a2 y + a3 y + a4 y + a5 y , f2∗ (y) = b0 + b1 y + b2 y + b3 y + b4 y + b5 y are given polynomial of degree, {aj }5j=0 , {bj }5j=0 Solution By looking for solutions to (2.5) in form of series in terms of orthogonal trigonometric polynomials, we reduce (2.5) into infinite system of linear equations We truncate the last system to N = with λ = 10 and we get approximate solutions to (2.5) In the concret case, where {a0 = 1, a1 = −2, a2 = 1, a3 = 0, a4 = −1, a5 = 1} v`a {b0 = 2, b1 = 1, b2 = −1, b3 = 4, b4 = 1, b5 = −2} the approximate solutions are following h i ∗ v1,6 (τ ) = √ − 0.92651 − 0.6436588τ + 2.10293τ − τ2 h i −√ 1.4994616τ − 0.50012τ − 0.999992τ + 1.000000τ − τ2 mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh mot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anhmot.so.lop.he.phuong.trinh.cap.va.ung.dung.ban.tom.tat.tieng.anh 23 and h i ∗ v2,6 (τ ) = √ 0.896727 − 2.3814487τ + 0.706384τ − τ2 h i + 1.500006τ − 4.999784τ + 1.0000048τ + 1.000000τ +√ − τ2 3.2.3 On convergence rate In this subsection we present convergence rate of approximate solutions vm,N (x) to prise one vm (x), m = 1, of the following singular integral equations system:  Z b Z   vm (t)dt X b  + vk (t)`mk (x − t)dt = −ifm (x), πi a x − t (3.3) a k=1    vm (t) ∈ L2 (a, b), m = 1, 2; a < x < b, ρ where −i `11 (x) = `22 (x) = π Z ∞ e−ξh sin(ξx)dξ, sinh(ξh) Z i ∞ sin(ξx) dξ `12 (x) = `21 (x) = π sinh(ξh) We have following Proposition on convergence rate for system (3.3) (k) Proposition 3.3 If the functions fm (x) pessess derivatives fm (x) that are continuous up to order k on [a, b] then   2 , m = 1, ||vm − vm,N ||L2ρ = ||φm (t) − φm,N (t)||L2−1 = O ρ N 2k−1 • Conclusions of the Chapter In this chapter we solve approximately a singular integral equationa system of the dual Fourier integral equations system, that arises in studying first mixed boundary value problem for harmonic equation in the Paragraph 2.1, by following steps - Reduce singular integral equations system into dimensionness form, then reduce it to infinite system of linear equations - Solve approximately the infinite system of linear equations by truncating it by N = v`a N = 7, then get approximate solutions for singular integral equations system - Present convergence rate for approximate solutions to prise ones 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