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

Abstract

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.

Computational Approach to Third-Order Nonlinear Boundary Value Problems via Efficient Decomposition Shooting Method

“…The absolute error difference between the exact analytical solution and the suggested solution E EDSM , which is further validated with E SRKM4 , is presented in Table 4. From Table 5, we can see that EDSM is the most accurate technique for solving the governing model in comparison with SRKM4 and the methods used in [33][34][35][36]. Furthermore, Figure 3 shows precise depictions of the analytical and approximate curves, where the two are in good conformity.…”

Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

“…From Table 4, we can see that EDSM is the most efficient method for solving example 2, compare with the results of the all methods in [23] [24] [25] [26] and SRKM4.…”

A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations

“…Collocation methods [7,8] have become increasingly popular for solving differential equations, since they are very useful in providing highly accurate solutions to nonlinear differential equations. Petrov-Galerkin method is widely used for solving ordinary and partial differential equations; see for example [9][10][11][12][13]. The Petrov-Galerkin methods [14] have generally come to be known as ''stablized'' formulations, because they prevent the spatial oscillations and sometimes yield nodally exact solutions where the classical Galerkin method would fail badly.…”