### Numerical solution of the linear inverse wave equation

Document Type : Research Paper

Authors

School of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364, Damghan, Iran.

Abstract

In this paper, a numerical method is proposed for the numerical solution of a linear wave equation with initial and boundary conditions by using the cubic B-spline method to determine the unknown boundary condition. We apply the cubic B-spline for the spatial variable and the derivatives, which generate an ill-posed linear system of equations. In this regard, to overcome, this drawback, we employ the Tikhonov regularization (TR) method for solving the resulting linear system. It is proved that the proposed method has the order of convergence $O\Bigl((\Delta t)^2+h^2\Bigr)$. Also, the conditional stability by using the Von-Neumann method is established under suitable assumptions. Finally, some numerical experiments are reported to show the efficiency and capability of the proposed method for solving inverse problems.

Keywords

[1] J.V. Beck, B. Blackwell, and C.R. St Clair Jr, Inverse heat conduction: III-posed problems, A Wiley-Interscience,
New York (1985).
[2] M. Bellassoued and M. Yamamoto, Inverse problems for wave equations on a riemannian manifold, Carleman
Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, 2017, pp. 111–166.
[3] J.M.G. Cabeza, J.A.M. Garc´ıa, and A.C. Rodr´ıguez, A sequential algorithm of inverse heat conduction problems
using singular value decomposition, Int. J. Thermal Sci. 44 (2005), no. 3, 235–244.
[4] M. Dehghan, S.A. Yousefi, and K. Rashedi, Ritz–Galerkin method for solving an inverse heat conduction problem
with a nonlinear source term via bernstein multi-scaling functions and cubic b-spline functions, Inv. Prob. Sci.
Engin. 21 (2013), no. 3, 500–523.
[5] S. Foadian, R. Pourgholi, and S. Hashem Tabasi, Cubic b-spline method for the solution of an inverse parabolic
system, Appl. Anal. 97 (2018), no. 3, 438–465.[6] J. Fritz, Partial differential equations, Springer-Verlag, 1982.
[7] J. Goh, A.A. Majid, and A.I.M. Ismail, Numerical method using cubic b-spline for the heat and wave equation,
Comput. Math. Appl. 62 (2011), no. 12, 4492–4498.
[8] C.L. Lawson and R.J. Hanson, Solving least squares problems, vol. 15, SIAM, 1995.
[9] Z. Lin and R.P. Gilbert, Numerical algorithm based on transmutation for solving inverse wave equation, Math.
Comput. Model. 39 (2004), no. 13, 1467–1476.
[10] L. Ma and Z. Wu, Radial basis functions method for parabolic inverse problem, Int. J. Comput. Math. 88 (2011),
no. 2, 384–395.
[11] R.C. Mittal and A. Tripathi, Numerical solutions of symmetric regularized long wave equations using collocation
of cubic b-splines finite element, Int. J. Comput. Meth. Engin. Sci. Mech. 16 (2015), no. 2, 142–150.
[12] P.M. Prenter, Splines and variational methods, John Wiley & Sons 537733048 (1975).
[13] G.D. Smith, Numerical solution of partial differential equations: Finite difference methods, Oxford university
press, 1985.
[14] A.N. Tikhonov and V.Y. Arsenin, Solutions of ill-posed problems, Washington, DC: VH Winston & Sons (1977).
[15] B. Wu and J. Liu, A global in time existence and uniqueness result for an integrodifferential hyperbolic inverse
problem with memory effect, J. Math. Anal. Appl. 373 (2011), no. 2, 585–604.
###### Volume 13, Issue 2July 2022Pages 1907-1926
• Receive Date: 08 September 2020
• Revise Date: 27 November 2020
• Accept Date: 26 December 2020