The thin plate spline collocation method for solving integro-differential equations arisen from the charged particle motion in oscillating magnetic fields

Purpose The purpose of this study is to obtain a scheme for the numerical solution of Volterra integro-differential equations with time periodic coefficients deduced from the charged particle motion for certain configurations of oscillating magnetic fields. Design/methodology/approach The method reduces the solution of these types of integro-differential equations to the solution of two-dimensional Volterra integral equations of the second kind. The new method uses the discrete collocation method together with thin plate splines constructed on a set of scattered points as a basis. Findings The scheme can be easily implemented on a computer and has a computationally attractive algorithm. Numerical examples are included to show the validity and efficiency of the new technique. Originality/value The author uses thin plate splines as a type of free-shape parameter radial basis functions which establish an effective and stable method to solve electromagnetic integro-differential equations. As the scheme does not need any background meshes, it can be identified as a meshless method.