Reconstructing an unknown potential term in the third-order pseudo-parabolic problem

• Published in: Computational & applied mathematics Vol. 40; no. 4
• Main Authors: Huntul, M. J., Dhiman, Neeraj, Tamsir, Mohammad
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied physics; Boundary conditions; Collocation methods; Computational mathematics; Computational Mathematics and Numerical Analysis; Inverse problems; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Noise sensitivity; Regularization; Stability analysis; Time dependence; Pseudo-parabolic equation; 35K70; Nonlinear optimization; Inverse identification problem; 65M30; Stability analysis; 65M22; Tikhonov regularization; CB-spline collocation method; 65M32
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-021-01532-4

The inverse problem of identifying the time-dependent potential term along with the temperature in a third-order pseudo-parabolic equation with initial and Neumann boundary conditions supplemented by the additional condition is, for the first time, numerically investigated. This problem emerges significantly in the modelling of various phenomena in physics and engineering. Although, the inverse problem is ill-posed by being sensitive to noise but has a unique solution. For the numerical realization, we apply the cubic B-spline (CB-spline) collocation method for discretizing the direct problem and the Tikhonov regularization for finding a stable and accurate solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB subroutine. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. The von Neumann stability analysis is also discussed.