Solutions of 2nd-order linear differential equations subject to Dirichlet boundary conditions in a Bernstein polynomial basis

H.M. Ahmed
2014 Journal of the Egyptian Mathematical Society  
KEYWORDS Fourier series in special orthogonal functions; Boundary value problems for ordinary differential equation; Initial-boundary value problems for second-order parabolic equations; Boundary value problems for non-linear first-order differential equations; Bernstein polynomial; Galerkin method Abstract An algorithm for approximating solutions to 2nd-order linear differential equations with polynomial coefficients in B-polynomials (Bernstein polynomial basis) subject to Dirichlet conditions
more » ... irichlet conditions is introduced. The algorithm expands the desired solution in terms of B-polynomials over a closed interval [0, 1] and then makes use of the orthonormal relation of B-polynomials with its dual basis to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of B-polynomials, and the procedure is much simpler compared to orthogonal polynomials for solving differential equations. The current procedure is implemented to solve five linear equations and one first-order nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations. MATHEMATICS SUBJECT CLASSIFICATION: 42C10; 35F30; 65L10; 65L60; 35K20 ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. ðb À xÞ nÀi ðx À aÞ i ðb À aÞ n ; i ¼ 0; 1; . . . ; n;
doi:10.1016/j.joems.2013.07.007 fatcat:mo2zvnqlhbhv7a6mall4jpngau