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Handbook of mathematics for engineers and scienteists part 122. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 16.2. Linear Integral Equations of the Second Kind with Variable Integration Limit 815 Let w w x be a solution of the simpler auxiliary equation with f x 1 and a 0 Aw x B J0 K x - t w t dt 1. 16.2.3.9 In this case the solution of the original equation 16.2.3.8 with an arbitrary right-hand side can be expressed via the solution of the auxiliary equation 16.2.3.9 by the formula d fx fx y x J w x - t f t dt f a w x - a J w x - t f t dt. 16.2.4. Construction of Solutions of Integral Equations with Special Right-Hand Side in this section we describe some approaches to the construction of solutions of integral equations with special right-hand sides. These approaches are based on the application of auxiliary solutions that depend on a free parameter. 16.2.4-1. General scheme. Consider a linear equation which we shall write in the following brief form L y fg x A 16.2.4.1 where L is a linear operator integral differential etc. that acts with respect to the variable x and is independent of the parameter A and fg x A is a given function that depends on the variable x and the parameter A. Suppose that the solution of equation 16.2.4.1 is known y y x A . 16.2.4.2 Let M be a linear operator integral differential etc. that acts with respect to the parameter A and is independent of the variable x. Consider the usual case in which M commutes with L. We apply the operator M to equation 16.2.4.1 and find that the equation L w fM x fM x M fg x A 16.2.4.3 has the solution w M y x A . 16.2.4.4 By choosing the operator M in a different way we can obtain solutions for other righthand sides of equation 16.2.4.1 . The original function fg x A is called the generating function for the operator L. 16.2.4-2. Generating function of exponential form. Consider a linear equation with exponential right-hand side L y eXx. 16.2.4.5 Suppose that the solution is known and is given by formula 16.2.4.2 . In Table 16.1 we present solutions of the equation L y f x with various right-hand sides these .
