An Efficient Optimization Approach for Solving Nonlinear Variable-Order Fractional PDEs With Nonlocal Boundary Conditions

Abstract

This paper presents an optimization algorithm designed to effectively handle a new general class of the nonlinear variable-order fractional partial differential equations (GCNV-OFPDEs) with nonlocal boundary conditions. Our approach involves utilizing a novel variant of the polynomials, namely generalized Abel polynomials (GAPs), and also new operational matrices to approximate the solution of the GCNV-OFPDEs. A key aspect of our algorithm is the transformation of GCNV-OFPDEs, along with their respective nonlocal boundary conditions, into systems of nonlinear algebraic equations. By solving these systems, we can determine the unknown coefficients and parameters. To address the nonlinear system, we employ the Lagrange multipliers to achieve optimal approximations. The convergence analysis of the approach is discussed. To validate the effectiveness of our algorithm, we conducted numerous experiments using various examples. The results obtained demonstrate the exceptional accuracy of our approach and its potential for extension to more complex problems in the future.