A Local Galerkin Integral Equation Method for Solving Integro-differential Equations Arising in Oscillating Magnetic Fields

The current work investigates a computational method to solve a class of integro-differential equations which describes the charged particle motion for certain configurations of oscillating magnetic fields. To handle the method, we first convert these types of integro-differential equations to two-dimensional Volterra integral equations of the second kind. Afterward, the solution of the mentioned Volterra integral equations can be estimated using the Galerkin method based on the moving least squares (MLS) approach constructed on scattered points. The MLS methodology is an effective technique for approximating an unknown function that involves a locally weighted least squares polynomial fitting. Since the scheme does not need any background meshes, it can be identified as a meshless method. The scheme is simple and effective to solve integro-differential equations and its algorithm can be easily implemented. Finally, numerical examples are included to show the validity and efficiency of the new technique.