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

Abstract

In this article, a new operational matrix method based on shifted Legendre polynomials is presented and analyzed for obtaining numerical spectral solutions of linear and nonlinear second-order boundary value problems. The method is novel and essentially based on reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations in the expansion coefficients of the sought-for spectral solutions. Linear differential equations are treated by applying the Petrov-Galerkin method, while the nonlinear equations are treated by applying the collocation method. Convergence analysis and some specific illustrative examples include singular, singularly perturbed and Bratu-type equations are considered to ascertain the validity, wide applicability and efficiency of the proposed method. The obtained numerical results are compared favorably with the analytical solutions and are more accurate than those discussed by some other existing techniques in the literature.

Modified Galerkin algorithm for solving multitype fractional differential equations

“…Corollary If γ = δ = 0, n = 1, and ν 2 = 2, then the derivative of the basis function is written as follows: This result is in complete agreement with the results obtained in the study of Abd‐Elhameed, Youssri, and Doha …”

Laguerre approach for solving system of linear Fredholm integro-differential equations

“…Solving the system of equations for N ¼ 3; 4; 5 by following the procedure stated above, yields the approximate solutions: u 1;3 ðsÞ ¼ À 1:9984 Â 10 À15 þ 1s À 8:88178 Â 10 À16 s 2 À 0:163805s 3 ; u 2;3 ðsÞ ¼ 1À3:33067 Â 10 À16 s À 0:5s 2 þ 0:0267372s 3 , and u 1;4 ðsÞ ¼ À 1:33227 Â 10 À15 þ 1s À 1:33227 Â 10 À14 s 2 À 0:193214s 3 þ 0:0392248s 4 ; u 2;4 ðsÞ ¼ 1 þ 8:52651 Â 10 À14 s À 0:5s 2 À 0:294265s 3 þ 0:620765s 4 , also u 1;5 ðsÞ ¼ 2:3892 Â 10 À13 þ 1s þ 4:01457 Â 10 À13 s 2 À 0:166728s 3 þ 0:000337574s 4 þ 0:00791926s 5 , u 2;5 ðsÞ ¼ 1: À 8:08242 Â 10 À13 s À 0:5s 2 À 0:00030213s 3 þ 0:0431559s 4 À 0:00244974s 5 :…”

“…There are many branches of science, such as control theory and financial mathematics, which leads to integro-differential equations (IDEs). In modern mathematics, IDEs mostly occur in many applied areas including engineering, physics and biology [1][2][3][4][5][6]. The resolution of many problems in physics and engineering leads to differential and integral equations in bounded or unbounded domains.…”

A numerical approach for a nonhomogeneous differential equation with variable delays

“…a. Delay differential equations of the type 1 arise in a variety of applications including control systems, electrodynamics, mixing liquids, neutron transportation, population models, physiological processes and conditions including production of blood cells [1,14,23,25,27,28,34,37].…”