An efficient numerical method based on Euler wavelets for solving nonlinear fractional order pantograph Volterra delay-integro-differential equations

Abstract The main purpose of this article is to solve the nonlinear pantograph Volterra delay integro-differential equation of fractional order. A numerical operational matrix approach based on Euler wavelets is proposed. For the proposed scheme, the fractional integral operational matrix is constructed. Then the nonlinear pantograph Volterra delay integro-differential equations are reduced to algebraic equations by using the fractional integral operational matrix. Several theorems are presented to establish the convergence and error analysis of the proposed method. To show the accuracy of the proposed technique, the numerical convergence rate has been shown. Additionally, some numerical problems are solved to justify the applicability and validity of the presented technique. Also, the numerical results have been documented graphically to describe the effectiveness of the approach. Furthermore, comparing numerical results with those obtained by known methods shows that the approach scheme is more efficient and accurate.