HANDBOOK OFINTEGRAL EQUATIONS phần 4 docx

... a special case of equation 3.8 .4 with g(x) = sinh(βx) and λ = µ/β. 9.  b a sinh 3  λ|x – t|  y(t) dt = f (x). Using the formula sinh 3 β = 1 4 sinh 3β – 3 4 sinh β, we arrive at an equation ... K(x, t). In the above equation, it is usually assumed that a ≤ x ≤ b. 4. 1. Equations Whose Kernels Contain Power-Law Functions 4. 1-1. Kernels Linear in the Arguments x and t 1. y(x)–λ ...

Ngày tải lên: 23/07/2014, 16:20

... x 4 = t 4 = 4 6 , x 5 = t 5 = 5 6 . Let us equate the determinant in the denominator of ( 14) with zero. After some algebraic manipulations, we obtain the following equation: 130µ 5 – 44 1µ 4 + 48 8µ 3 – ... Riemann problem (44 ) in the usual form (see Subsection 10 .4- 4) Ξ + (u)=D(u)Ξ – (u)+H(u), –∞ < u < ∞, (45 ) where D(u)= 1+K 2 (u) 1+K 1 (u) , H(u)= K 2 (u) – K 1 (u) 1+K...

Ngày tải lên: 23/07/2014, 16:20

... equation 6.8 .45 with f(t, y)=A tanh(βy). 43 . y(x)+A  b a coth[βy(x)] coth[γy(t)] dt = h(x). This is a special case of equation 6.8 .43 with g(x, y)=A coth(βy) and f(t, y) = coth(γy). 44 . y(x)+A  b a y(xt) ... equation 6.8 .45 with f(t, y)=A sinh(βy). 41 . y(x)+A  b a tanh[βy(x)] tanh[γy(t)] dt = h(x). This is a special case of equation 6.8 .43 with g(x, y)=A tanh(βy) and f(t, y) =...

Ngày tải lên: 23/07/2014, 16:20

... sin(ax)  47 1 p 4 – a 4 1 2a 3  sinh(ax) – sin(ax)  48 p p 4 – a 4 1 2a 2  cosh(ax) – cos(ax)  49 p 2 p 4 – a 4 1 2a  sinh(ax) + sin(ax)  50 p 3 p 4 – a 4 1 2  cosh(ax) + cos(ax)  51 1 p 4 + ... a) e –ax + 1 b 2 (a – b) e –bx 43 1 (p + a) 2 (p + b) 2 1 (a – b) 2  e –ax  x + 2 a – b  + e –bx  x – 2 a – b  44 1 (p + a) 4 1 6 x 3 e –ax 45 p (p + a)...

Ngày tải lên: 23/07/2014, 16:20

... Press LLC Chapter 14 Methods for Solving Nonlinear Integral Equations 14. 1. Some Definitions and Remarks 14. 1-1. Nonlinear Volterra Integral Equations Nonlinear Volterra integral equations can be ... then y 1 (x)=  x 0 (–1) dt = –x, y 2 (x)=  x 0 (t 3 – 1) dt = –x + 1 4 x 4 , y 3 (x)=  x 0  t  1 16 t 8 – 1 2 t 5 + t 2  – 1  dt = –x + 1 4 x 4 – 1 14 x 7 + 1 160 x 10 ....

Ngày tải lên: 23/07/2014, 16:20

... transforms enables one to reduce some integral equations on the entire axis and on the semiaxis to algebraic equations for transforms. These algebraic equations can readily be solved for the transform ... ∞, (1) where f(x), y(x) ∈ L 2 (–∞, ∞) and K(x) ∈ L 1 (–∞, ∞). Page 49 4 © 1998 by CRC Press LLC © 1998 by CRC Press LLC TABLE 4 Solutions of the equation L [y]=f(x) with generating...

Ngày tải lên: 23/07/2014, 16:20

... equation 4. 9.10 with h(x) = tan(βx). 4. 5 -4. Kernels Containing Cotangent 44 . y(x) – λ  b a cot(βx)y(t) dt = f (x). This is a special case of equation 4. 9.1 with g(x) = cot(βx) and h(t)=1. 45 . y(x) ... equation 4. 9.1 with g(x) = tan k (βx) and h(t) = tan m (µt). 40 . y(x) – λ  b a t k tan m (βx)y(t) dt = f (x). This is a special case of equation 4. 9.1 with g(x) = tan m (βx)...

Ngày tải lên: 23/07/2014, 16:20

... the form y(x)= m  k=1 A k B k exp(–λ k x), B k =1– a  π λ k  ln (4 k )+C  , where C = 0.5772 is the Euler constant. 2.5. Equations Whose Kernels Contain Trigonometric Functions 2.5-1. Kernels ... f(x). The substitution u(x)=e –λx y(x) leads to an equation of the form 2.1 .49 : u(x)+A  x a (x – t)t k u(t) dt = f (x)e –λx . 34. y(x)+A  x a (x k – t k )e λ(x–t) y(t) dt = f(x). The sub...

Ngày tải lên: 23/07/2014, 16:20

... – t)]w(t) dt = e –µx f(x). 4.  x a e µ(x–t) cosh 4 [λ(x – t)]y(t) dt = f(x). The substitution w(x)=e –µx y(x) leads to an equation of the form 1.3.19:  x a cosh 4 [λ(x – t)]w(t) dt = e –µx f(x). Page ... t  f(t) dt. 63.  x a (x – t) 1 /4 I 1/2  λ √ x – t  y(t) dt = f(x). Solution: y(x)=  2 πλ d 2 dx 2  x a cos  λ √ x – t  √ x – t f(t) dt. 64.  x a (x – t) 3 /4 I 3/2 ...

Ngày tải lên: 23/07/2014, 16:20

... Integral Equations 14. 1. Some Definitions and Remarks 14. 1-1. Nonlinear Volterra Integral Equations 14. 1-2. Nonlinear Equations With Constant Integration Limits 14. 2. Nonlinear Volterra Integral Equations 14. 2-1. ... Integral Equations 13 .4- 1. Certain Properties of Singular Operators 13 .4- 2. The Regularizer 13 .4- 3. The Methods of Left and Right Regularization 13 .4- 4. T...

Ngày tải lên: 23/07/2014, 16:20