derivation of finite difference equations

... Accuracy Schemes 3.8 Accuracy of Finite Difference Solutions 3.9 Summary References Solution Methods of Finite Difference Equations 4.1 Elliptic Equations 4.1.1 Finite Difference Formulations 4.1.2 ... FINITE DIFFERENCE METHODS Derivation of Finite Difference Equations 3.1 Simple Methods 3.2 General Methods 3.3 Higher Order Derivatives 45 45 46 50 v vi CONTENTS 3.4 Multidimensional Finite Difference ... Governing Equations 6.1.2 Subsonic Potential Flows 6.1.3 Transonic Potential Flows 6.2 Euler Equations 6.2.1 Mathematical Properties of Euler Equations 6.2.1.1 Quasilinearization of Euler Equations...

Ngày tải lên: 01/04/2014, 10:45

... [2–7] which deal with existence results for this kind of problems For the qualitative study of this type of problems we can cite the survey of Hartung et al [8] and references therein As main improvements ... and compact set of fixed points in C(I), which are exactly the solutions of problem (4) Claim 2: Every solution x of (4) satisfies α ≤ x ≤ β on I and, therefore, it is a solution of (1) in [α, β] ... that all solutions x of (4) obey x ≤ β on I Claim 3: The set of solutions of problem (1) in [α, β] has maximal and minimal elements The set S = {x ∈ C(I) : x is a solution of (1) , x ∈ [α, β]}...

Ngày tải lên: 20/06/2014, 20:20

... delay dynamic equations of first order,” Advances in Difference Equations, vol 2008, Article ID 458687, 12 pages, 2008 10 H A Agwo, “On the oscillation of first order delay dynamic equations with ... functions are assumed to vanish out of their domains, ϕ αi t t0 , ∞ T if αi t ≥ t0 for t ∈ Advances in Difference Equations Proof As the uniqueness for the solution of 2.1 was proven in , it suffices ... solutions of 4.11 and 4.13 illustrates the result of Theorem 4.5, √ here y t > x t for all t ∈ 3, ∞ √N Corollary 4.8 Let x be a solution of 3.1 , and Y ·, s > on s, ∞ fundamental solution of xΔ...

Ngày tải lên: 21/06/2014, 11:20

... see [1, 2], that the existence of α and β, a pair of lower and upper solutions of problem (PN ), such that α ≤ β in JN , does not imply the existence of a solution of this problem Alberto Cabada ... more precise description of the set of solutions of problem (PN ) Lemma 2.5 Assume conditions (H) and (H f ) Let u, v ∈ [α,β] be two solutions of problem (PN ) Then, one of the following statements ... following characterization of the set of solutions of problem (PN ) Corollary 3.2 Let u : JN → R, then u is a solution of problem (PN ) if and only if u is a fixed point of operator T Proposition...

Ngày tải lên: 23/06/2014, 00:20

... Advances in Difference Equations 2011, 2011:14 http://www.advancesindifferenceequations.com/content/2011/1/14 Page of 17 It completes the proof Now consider the existence of solutions of PBVP (1.1) ... (2:6) Xing and Fu Advances in Difference Equations 2011, 2011:14 http://www.advancesindifferenceequations.com/content/2011/1/14 Page of 17 is equivalent to the family of PBVPs x + m(t)x = λg(t, ... 0.2 Figure Figure of Example 2.2 0.4 0.6 0.8 Xing and Fu Advances in Difference Equations 2011, 2011:14 http://www.advancesindifferenceequations.com/content/2011/1/14 Page of 17 First note that...

Ngày tải lên: 21/06/2014, 02:20

... differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of differential equations without impulse effects Moreover, such equations ... boundary value problems of impulsive dynamic equations on time scales [22–36] However, to the best of our knowledge, few papers concerning PBVPs of impulsive dynamic equations on time scales ... first-order impulsive dynamic equations on time scales Da-Bin Wang Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China Email address:...

Ngày tải lên: 21/06/2014, 17:20

... extremal solutions of 1.1 are given as the infimum of all upper solutions and the supremum of all lower solutions, that is, the least solution of 1.1 is given by inf u t : u upper solution of 1.1 , t ... family of admissible non-quasisemicontinuity curves contained in E Then the conclusions of Theorem 3.1 hold true 12 Boundary Value Problems Proof (Outline) It follows the same steps as the proof of ... Solutions Another modification of the concepts of lower and upper solutions concerns the possibility of allowing jumps in their graphs Since the task of finding a pair of lower and upper solutions...

Ngày tải lên: 21/06/2014, 20:20

... difference equations with impulses,” International Journal of Difference Equations, vol 1, no 2, pp 225–239, 2006 15 Y Li, X Fan, and L Zhao, “Positive periodic solutions of functional differential equations ... emerging as an important area of investigation, since it is a lot richer than the corresponding theory of differential equations without impulse effects Moreover, such equations may exhibit several ... problems for impulsive differential equations and impulsive difference equations have received much attention 9–19 On the other hand, recently, the theory of dynamic equations on time scales has become...

Ngày tải lên: 21/06/2014, 20:20

... to the existence of the nontrivial solution x of the homogeneous problem Mx 0, lx Theorem 4.15 Conditions of each of Theorems 3.1–3.6 imply assertions 1–4 of Theorem 4.2 for equations 3.3 , 3.5 ... Denote by C0 t, s the Cauchy function of the equation Nx fundamental function of 3.1 0, which is also the 18 Journal of Inequalities and Applications Proofs of Theorems 3.1–3.6 are based on the ... is less than one f is nondecreasing and the norm of the operator S : L∞ 0,∞ 0,∞ Proof of Lemma 3.8 Assertions about nonnegativity of solution x of the homogeneous equation Mx and its derivative...

Ngày tải lên: 22/06/2014, 03:20

... there exists n0 ∈ N such that 2.11 holds, then every solution of 1.9 oscillates on t0 , ∞ T Proof The proof is an immediate consequence of Lemmas 2.1 and 2.2 We need the following lemmas in the ... and 2.11 hold Then, every solution of 1.9 is oscillatory on t0 , ∞ T Proof The proof follows from Lemmas 2.1, 2.2, and 2.5 Remark 2.7 We obtain the main results of 7, by letting n0 in Theorem 2.6 ... of delay differential equations, ” Journal of Mathematical Analysis and Applications, vol 219, no 1, pp 165–178, 1998 J Shen and X Tang, “New oscillation criteria for linear delay differential equations, ”...

Ngày tải lên: 22/06/2014, 11:20

... that the results of this paper are presented in a form which is essentially new and of high degree of generality Also, for related results in oscillation of forced differential equations, we refer ... (n)xμ (n) − h(m,n)x(n) n =k The rest of the proof is similar to that of Theorem 2.2 and hence omitted Also, we discuss the oscillatory behavior of a special case of (4.7), namely, the equation Δ2 ... n =k (4.18) The rest of the proof is similar to that of Theorem 3.1 and hence omitted Remark 4.7 If we let λ = in (4.16), we see that the inequality (4.18) in the proof of Theorem 4.8 reduces...

Ngày tải lên: 22/06/2014, 22:20

... solutions of (2.1)–(2.3) with α(t) ≤ β(t) for all t ∈ J Then, there exists a solution u to (2.1)– (2.3) such that u ∈ [α,β] F M Atici and D C Biles 123 Proof Our proof follows that of Cabada ... u(0) ≡ u(0) − B(u(0),u(T)) Claim 2.3 If u is a fixed point of the operator G, then u is a solution of (2.1)–(2.3) such that u ∈ [α,β] Proof of Claim 2.3 We assume that u ∈ A satisfies u(t) = p 0,u(0) ... respectively, of the problem (3.1)–(3.5) such that α ≤ β on T Then every solution of the BVP (3.11)–(3.15) belongs to the sector [α,β] F M Atici and D C Biles 129 Proof Let y be a solution of (3.11)–(3.15),...

Ngày tải lên: 23/06/2014, 00:20

... than those of Wolff 1.3 Overview of the paper The next two sections of this paper are concerned with reducing the proof of Theorem 1.2 to establishing an existence result for smooth data of small ... the case of smooth, compactly supported data of small norm, namely Proposition 3.1 In Section we present the proof of Proposition 3.1 by the continuity method At the heart of this proof is a recursive ... combination of three terms, each of which takes the form of an operator of type S1,1 acting on du, where any given seminorm of the symbol is bounded by c(m) du L∞ , with c(m) an appropriate power of m.2...

Ngày tải lên: 15/03/2014, 09:20

... the ideal of smoothing operators The topology of Ψbc (X) is given in terms of the conormal seminorms of the Schwartz kernel K of its elements; these seminorms can be stated in terms of the Besov ... quotient of H −1 (X) by the annihilator of H0 (X) In terms of the identification spaces in the penultimate paragraph of Remark 3.1, H −1 (X) = of the H loc ¯ −1 Hloc (X ◦ ) in the notation of [4, ... C 781 PROPAGATION OF SINGULARITIES a neighborhood of WFb (A), which is possible in view of (ii) at the beginning of the proof Then the second integral on the left-hand side of (4.6) can be ˜...

Ngày tải lên: 15/03/2014, 09:20

Ngày tải lên: 17/03/2014, 14:44

... generator of the satisfying principal filter of the conjunction of such constraints is the unification of the generators of the satisfying principal filters of each of the conjuncts Thus the set of ... restricted to conjunctions of equality constraints (i.e requirements that the value of a path of attributes is equal to a constant or the value of another path) then the set of satisfying attributevalue ... s of '=' d e n o t e identical elements distinct f r o m the d e n o t a t i o n of_ L Even t h o u g h there are infinitely m a n y instances of the s c h e m a in (A5) (since there are infinitely...

Ngày tải lên: 24/03/2014, 02:20

... irrespective of the fact that this sequence is periodic or not, some state will repeat finitely often whereas other will repeat infinitely often The (finite) set of states occurring infinitely often in ... point of view of finite word instead of infinite word reading automata, as for instance in [6,10,11,12,13,14,15] Unfortunately, as opposed to the case of infinite words, the classification theory of ... to the attractive properties of the considered networks, and hence provides a new refined measurement of the computational capabilities of these networks in terms of their attractive behaviours...

Ngày tải lên: 28/04/2014, 10:06

... - w ) P = c2 dx/dt =w/k = kdx/dt –w =0 angular frequency pi k = w k ==2 w c c Wavefront = Line of constant phase Wavelength=shortest distance between adjacent peaks 2D Plane Wave Solution Phase ... Ae = Ae i(k x + k z - (2) wt) x z i(k r - wt) k = (k , k z ) x (2) = |k|(sin , cos ) k Equation of a line: k r = cnst k is Perpendicular to Wavefront z r k k = |r||k| cos(O) = constant Any pt ... z - (2) wt) x z i(k r - wt) k = (k , k z ) x x= sinO z (2) = |k|(sin , cos ) k z= cosO Equation of a line: k r = cnst = 2pi/|k| Time (s) Apparent Velocity dx/dt=apparent V x = x dz/dt=apparent...

Ngày tải lên: 28/04/2014, 14:16

... a class of higher-order difference equations Huantao Zhu1 and Weibing Wang∗2 Hunan College of Information, Changsha, Hunan 410200, P.R China Department of Mathematics, Hunan University of Science ... and multiple positive periodic solutions of (1.3) Our aim of this article is to study the existence of periodic solutions for the higherorder difference equations (1.1) using the well-known Schauder’s ... −h −1 p−i (Hp x)(n + (i − 1)k), x ∈ X − 1) i=1 Clearly, the fixed point of Tp in X is a periodic solution of (1.1) Proof of Theorem 1.1 Let p = r and Ω = {x ∈ X : m ≤ x(n) ≤ M for n ∈ Z}, then...

Ngày tải lên: 18/06/2014, 22:20