HANDBOOK OFINTEGRAL EQUATIONS phần 5 ppsx

... f(a)+f(b). (5) Note a useful consequence of (5) , y  x (a)+y  x (b)=f  x (a)+f  x (b), (6) which can be used together with one of conditions (5) . Equation (3) under the boundary conditions (5) determines ... a)]ϕ(a)=–λϕ(b); ϕ(x)=y(x) – f(x). (5) Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to...

Ngày tải lên: 23/07/2014, 16:20

... a < u 15 K 0  a √ x 2 + b 2  , a, b >0 π 2 √ u 2 + a 2 exp  –b √ u 2 + a 2  • References for Supplement 6: G. Doetsch (1 950 , 1 956 , 1 958 ), H. Bateman and A. Erd ´ elyi (1 954 ), V. A. ... + sin(ax)  50 p 3 p 4 – a 4 1 2  cosh(ax) + cos(ax)  51 1 p 4 + a 4 1 a 3 √ 2  cosh ξ sin ξ – sinh ξ cos ξ  , ξ = ax √ 2 52 p p 4 + a 4 1 a 2 sin  ax √ 2  sinh  ax √ 2  53...

Ngày tải lên: 23/07/2014, 16:20

... Complete Singular Integral Equations Solvable in a Closed Form In contrast with characteristic equations and their transposed equations, complete singular integral equations cannot be solved ... equation ϕ(t)+R 1   L K(t, τ)ϕ(τ) dτ  = R 1 [f(t)] + b(t)Z(t)P ν–p–1 (t), ( 45) Page 655 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 12.4. Singular Integral Equations of the First Ki...

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... differential equations of order 2n with constant coefficients (see equations 4 .5. 29 and 4 .5. 32 in the first part of the book). • References for Section 11.12: W. B. Davenport and W. L. Root (1 958 ), I. ... choose equidistant points x i and t j and take n = 5. This implies x 1 = t 1 = 1 6 , x 2 = t 2 = 2 6 , x 3 = t 3 = 3 6 , x 4 = t 4 = 4 6 , x 5 = t 5 = 5 6 . Let us equate the...

Ngày tải lên: 23/07/2014, 16:20

... s), Page 4 95 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 9 .5- 2. A Generating Function of Exponential Form Consider a linear equation with exponential right-hand side L [y]=e λx . (5) Suppose ... known. Below we list some classes of integral equations that can be reduced to ordinary differential equations with constant coefficients. 9.7-1. Equations With Kernel Containing a Sum o...

Ngày tải lên: 23/07/2014, 16:20

... CRC Press LLC 55 . y(x)+  b a y(x – t)f  t, y(t)  dt = Ax + B. A solution: y(x)=px + q, (1) where p and q are roots of the following system of algebraic (or transcendental) equations: p + ... ϕ(x)=y(x) – g(x). (5) Equation (3) under the boundary conditions (5) determines the solution of the original integral equation (there may be several solutions). Conditions (5) make it possible .....

Ngày tải lên: 23/07/2014, 16:20

... V. Widder (1 955 ), V. A. Ditkin and A. P. Prudnikov (19 65) . 16.  ∞ 0 K(x – t)y(t) dt = f (x). The Wiener–Hopf equation of the first kind. This equation is discussed in Subsection 10 .5- 1 in detail. 3.8-4. ... ξ), we arrive at an integral equation with difference kernel of the form 3.8. 15:  ∞ –∞ G(z – τ)w(τ) dτ = F (z). 3.8 -5. Equations of the Form  b a K(x, t)y(···) dt = F (x) 26....

Ngày tải lên: 23/07/2014, 16:20

... form y(x)= m  k=1 A k B k exp(–λ k x), B k =1– a  π λ k  ln(4λ k )+C  , where C = 0 .57 72 is the Euler constant. 2 .5. Equations Whose Kernels Contain Trigonometric Functions 2 .5- 1. Kernels Containing Cosine 1. y(x) – A  x a cos(λx)y(t) ... f(x). The substitution u(x)=e –λx y(x) leads to an equation of the form 2.1 .52 : u(x)+A  x a (x k – t k )u(t) dt = f (x)e –λx . 35. y(x) – λ...

Ngày tải lên: 23/07/2014, 16:20

... 1 .5. 24:  x a (cos λ x – cos λ t)w(t) dt = e –µx f(x). 56 .  x a e µ(x–t)  A cos λ x + B cos λ t  y(t) dt = f(x). The substitution w(x)=e –µx y(x) leads to an equation of the form 1 .5. 25:  x a  A ... cos x . 54 .  x a e µ(x–t) (cos t – cos x) λ y(t) dt = f(x), 0 < λ <1. Solution: y(x)=ke µx sin x  1 sin x d dx  2  x a e –µt sin tf(t) dt (cos t – cos x) λ , k = sin(πλ)...

Ngày tải lên: 23/07/2014, 16:20

... Complicated Argument 3 .5. Equations Whose Kernels Contain Trigonometric Functions 3 .5- 1. Kernels Containing Cosine 3 .5- 2. Kernels Containing Sine 3 .5- 3. Kernels Containing Tangent 3 .5- 4. Kernels Containing ... Cotangent 3 .5- 5. Kernels Containing a Combination of Trigonometric Functions 3 .5- 6. Equations Containing the Unknown Function of a Complicated Argument 3 .5- 7. A Singu...

Ngày tải lên: 23/07/2014, 16:20