A spectral method based on Bernstein orthonormal basis functions for solving an inverse Roseneau equation

• Published in: Computational & applied mathematics Vol. 41; no. 5
• Main Author: Rashedi, Kamal
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied physics; Basis functions; Boundary conditions; Computational mathematics; Computational Mathematics and Numerical Analysis; Inverse problems; Mathematical analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Smooth boundaries; Spectral methods; Inverse Rosenau equation; Spectral method; 65N35; 35R30; Satisfier function; 65M30; 35R25; Operational matrices
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-022-01908-0

In this work, we study an inverse problem with unknown boundary conditions in the one-dimensional Rosenau equation. It is assumed that we have extra information such as some interior measurements to compensate the insufficiency of the input information. In the first step, by applying the satisfier function, the main problem is reformulated as an equivalent problem with homogeneous initial and boundary conditions. For the contaminated measurements, we apply the mollification method to obtain stable numerical derivatives and produce smooth boundary data. Then, we employ a direct technique based on the operational matrices of integration and differentiation of the orthonormal Bernstein basis functions together with the collocation technique to reduce the main problem to the solution of a system of algebraic equations. Numerical simulations for three test examples are presented to show the accuracy of the proposed solution.