Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL EQUATIONSWe shall illustrate the principles involved by considering the differential equationy ′′ (x) =f(x, y), (23.1)where f(x, y) can be any function of x and y but not of y ′ (x). Equation (23.1)thus represents a large class of linear and non-linear second-order differentialequations.We can convert (23.1) into the corresponding integral equation by first integratingwith respect to x to obtainy ′ (x) =Integrating once more, we findy(x) =∫ x0du∫ x0∫ u0f(z,y(z)) dz + c 1 .f(z,y(z)) dz + c 1 x + c 2 .Provided we do not change the region in the uz-plane over which the doubleintegral is taken, we can reverse the order of the two integrations. Changing theintegration limits appropriately, we findy(x) ==∫ x0∫ x0f(z,y(z)) dz∫ xzdu + c 1 x + c 2 (23.2)(x − z)f(z,y(z)) dz + c 1 x + c 2 ; (23.3)this is a non-linear (for general f(x, y)) Volterra integral equation.It is straightforward to incorporate any boundary conditions on the solutiony(x) by fixing the constants c 1 and c 2 in (23.3). For example, we might have theone-point boundary condition y(0) = a and y ′ (0) = b, for which it is clear thatwe must set c 1 = b and c 2 = a.23.2 Types of integral equationFrom (23.3), we can see that even a relatively simple differential equation suchas (23.1) can lead to a corresponding integral equation that is non-linear. In thischapter, however, we will restrict our attention to linear integral equations, whichhave the general formg(x)y(x) =f(x)+λ∫ baK(x, z)y(z) dz. (23.4)In (23.4), y(x) is the unknown function, while the functions f(x), g(x) andK(x, z)are assumed known. K(x, z) is called the kernel of the integral equation. Theintegration limits a and b are also assumed known, and may be constants orfunctions of x, andλ is a known constant or parameter.804