Communications in Mathematical Sciences

Volume 15 (2017)

Number 3

A time integration method of approximate fundamental solutions for nonlinear Poisson-type boundary value problems

Pages: 693 – 710



Corey Jones (Department of Mathematics, Jones County Junior College, Ellisville, Mississippi, U.S.A.)

Haiyan Tian (Department of Mathematics, University of Southern Mississippi, Hattiesburg, Ms., U.S.A.)


A time-dependent method is coupled with the method of approximate particular solutions (MAPS) and the method of approximate fundamental solutions (MAFS) of Delta-shaped basis functions to solve a nonlinear Poisson-type boundary value problem on an irregular shaped domain. The problem is first converted into a sequence of time-dependent nonhomogeneous modified Helmholtz boundary value problems through a fictitious time integration method. Then the superposition principle is applied to split the numerical solution at each time step into an approximate particular solution and a homogeneous solution. A Delta-shaped basis function is used to provide an approximation of the source function at each time step. This allows for an easy derivation of an approximate particular solution. The corresponding homogeneous boundary value problem is solved using MAFS, and also with the method of fundamental solutions (MFS) for comparison purposes. Numerical results support the accuracy and validity of this computational method.


fictitious time, nonlinear Poisson-type equations, Delta-shaped basis, modified Helmholtz equation, approximate particular solutions, approximate fundamental solutions

2010 Mathematics Subject Classification

65N35, 65N80

Published 24 February 2017