A novel differential-integral quadrature method for the solution of nonlinear integro-differential equations

In this work, we introduce an integration matrix operator that is fully consistent with the differentiation matrix operator defined by the differential quadrature method (DQM). Using these operators, a generic differential-integral quadrature method DIQM is proposed to directly discretize an integro-differential equation as a system of algebraic equations. To extend the applicability of the proposed DIQM to solve nonlinear and/or variable coefficients integro-differential equation, some matrix manipulations are introduced. A stability analysis for Volterra integro-partial-differential equations is presented and the exponential convergence of the proposed method is examined. Various types of integro-differential equations are solved including ordinary/partial, linear/nonlinear, Volterra parabolic/hyperbolic integro-differential equations with different boundary and initial conditions. Numerical results demonstrate the exponential convergence and the applicability of DIQM.

Automated summary

An integration matrix operator is introduced in this work to discretize integro-differential equations, along with a generic differential-integral quadrature method (DIQM). Stability analysis and numerical results demonstrate the exponential convergence and applicability of the proposed method.