Power series solution for integral equations
Power series solution for integral equations [ edit ] See also: Liouville–Neumann series In many cases, if the Kernel of the integral equation is of the form K ( xt ) and the Mellin transform of K ( t ) exists, we can find the solution of the integral equation � ( � ) = � ∫ 0 ∞ � ( � � ) � ( � ) � � in the form of a power series � ( � ) = ∑ � = 0 ∞ � � � ( � + 1 ) � � where � ( � ) = ∑ � = 0 ∞ � � � − � , � ( � + 1 ) = ∫ 0 ∞ � ( � ) � � � � are the Z -transform of the function g ( s ) , and M ( n + 1) is the Mellin transform of the Kernel. Numerical solution [ edit ] It is worth noting that integral equations often do not have an analytical solution, and must be solved numerically. An example of this is evaluating the electric-field integral equation (EFIE) or magnetic-field integral equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem. One method ...