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The major purpose of this article is to expand the three-dimensional (3D) modified moving least-square method for the numerical solution of 3D linear and nonlinear Volterra–Fredholm integral equations of the second kind. This approach is very convenient for solving integral equations in high dimensions and it does not require any need for mesh connectivity. The size of the support used is the only factor that has a significant effect on the maximum errors of the MLS method. To overcome this problem, the MMLS approach with a non-singular moment matrix is applied to obtain better results than MLS approximation on using the best support, then applying the method in three dimensions can be easily achieved. The numerical experiments of the MMLS and classical MLS techniques are presented to show the difference between both methods for multi-dimensional problems. The convergence analysis is provided and some numerical tests are given to prove the applicability of this technique.

Numerical solution for three-dimensional nonlinear mixed Volterra–Fredholm integral equations via modified moving least-square method