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In this paper, we used the three-dimensional triangular functions (3D-TFs) for the numerical solution of three-dimensional nonlinear mixed Volterra–Fredholm integral equations. First, 3D-TFs and their properties are described. Then the properties of 3D-TFs together with their operational matrix are used to reduce the problem to a nonlinear system of algebraic equations. Furthermore, existence and uniqueness of the solution of three-dimensional nonlinear mixed Volterra–Fredholm integral equations are proved. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique. Also, some interesting comparisons between proposed method, block-pulse functions (BPFs) method and modified block-pulse functions (MBPFs) method are presented

Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra–Fredholm integral equations