Numerical solutions of nonlinear two-dimensional partial Volterra integro-differential equations by Haar wavelet

A non-linear kernel comprising a function of partial derivatives of arbitrary order is approximated by Theorem 2 in Ref. [12]. After substituting the kernel approximation in the original equation, a nonlinear system is obtained using the Haar wavelet. Solving the nonlinear system, the nonlinear two-dimensional integro-differential Volterra equation with partial derivatives is converted to a simple equation containing partial derivatives. Solving this simple equation, we can approximate the solution of the nonlinear two-dimensional integro-differential Volterra equation.