Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.

An Accurate Spectral Galerkin Method for Solving Multiterm Fractional Differential Equations

This paper reports a new formula expressing the Caputo fractional derivatives for any order of shifted generalized Jacobi polynomials of any degree in terms of shifted generalized Jacobi polynomials themselves. A direct solution technique is presented for solving multiterm fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using spectral shifted generalized Jacobi Galerkin method. The homogeneous initial conditions are satisfied exactly by using a class of shifted generalized Jacobi polynomials as a polynomial basis of the truncated expansion for the approximate solution. The approximation of the spatial Caputo fractional order derivatives is expanded in terms of a class of shifted generalized Jacobi polynomialsJnα,−β(x)withx∈(0,1), andnis the polynomial degree. Several numerical examples with comparisons with the exact solutions are given to confirm the reliability of the proposed method for multiterm FDEs.