An inverse problem of fourth-order partial differential equation with nonlocal integral condition
Tóm tắt
In this study, the time-dependent potential coefficient in a higher-order PDE with initial and boundary conditions is numerically constructed for the first time from a nonlocal integral condition. Even though the inverse identification problem investigated in this study is ill-posed, it has a unique solution. For discretizing the direct problem and finding stable and accurate solutions, we employ the Quintic B-spline (QBS) collocation and Tikhonov regularization methods, respectively. The following nonlinear minimization problem is solved using MATLAB. The collected findings demonstrate that accurate and stable solutions can be found.
Từ khóa
Tài liệu tham khảo
Abbasova, K.E., Mehraliyev, Y.T., Azizbayov, E.I.: Inverse boundary-value problem for linearized equation of motion of a homogeneous elastic beam. Int. J. Appl. Comput. Math. 33, 157–170 (2020)
Caglar, H.N., Caglar, S.H., Twizell, E.H.: The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions. Appl. Math. Lett. 12, 25–30 (1999)
De Boor, C.: On the convergence of odd-degree spline interpolation. J. Approx. Theory 1(4), 452–463 (1968)
Dhiman, N., Tamsir, M.: Re-modified quintic B-spline collocation method for the solution of Kuramoto–Sivashinsky type equations. Multidiscip. Model. Mater. Struct. (2018). https://doi.org/10.1108/MMMS-06-2018-0111
DuChateau, P., Zachmann, D.: Applied Partial Differential Equations. Harper & Row, New York (1989)
Gebremedhin, G.S., Jena, S.R.: Approximate solution of ordinary differential equation via hybrid block approach. Int. J. Emerg. Technol. 10, 201–211 (2019)
Gebremedhin, G.S., Jena, S.R.: Approximate solution of a fourth order ordinary differential equation via tenth step block method. Int. J. Comput. Sci. Math. 11, 253–262 (2020)
Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven (2003)
Huntul, M., Tamsir, M.: Identifying an unknown potential term in the fourth-order Boussinesq–Love equation from mass measurement. Eng. Comput. (2021). https://doi.org/10.1108/EC-12-2020-0757
Huntul, M.J., Tamisr, M., Ahmadini, A.: An inverse problem of determining the time-dependent potential in the higher-order Boussinesq–Love equation from boundary data. Eng. Comput. (2021). https://doi.org/10.1108/EC-08-2020-0459
Huntul, M.J., Tamsir, M., Dhiman, N.: An inverse problem of identifying the time-dependent potential in a fourth-order pseudo-parabolic equation from additional condition. Numer. Methods Partial Differ. Equ. (2021). https://doi.org/10.1002/num.22778
Ivanov, V.K.: On linear problems which are not well-posed. Dokl. Akad. Nauk SSSR 145, 270–272 (1962)
Jena, S.R., Gebremedhin, G.S.: Approximate solution of a fifth order ordinary differential equation with block method. Int. J. Comput. Sci. Math. 12, 413–426 (2020)
Jena, S.R., Gebremedhin, G.S.: Numerical treatment of Kuramoto–Sivashinsky equation on B-spline collocation. Arab J. Basic Appl. Sci. 28, 283–291 (2021)
Jena, S.R., Gebremedhin, G.S.: Decatic B-spline collocation scheme for approximate solution of Burgers’ equation. Numer. Methods Partial Differ. Equ. (2021). https://doi.org/10.1002/num.22747
Jena, S.R., Gebremedhin, G.S.: Computational technique for heat and advection–diffusion equations. Soft Comput. 25, 1139–1150 (2021)
Jena, S.R., Mohanty, M.: Numerical treatment of ODE (fifth order). Int. J. Emerg. Technol. 10, 191–196 (2019)
Jena, S.R., Mohanty, M., Mishra, S.K.: Ninth step block method for numerical solution of a fourth order ordinary differential equation. Adv. Model. Anal. A 55, 47–56 (2018)
Jena, S.R., Nayak, D., Acharya, M.M.: Application of mixed quadrature rule on electromagnetic field problems. Comput. Math. Model. 28, 267–277 (2017)
Jena, S.R., Senapati, A., Gebremedhin, G.S.: Numerical study of solitions in BFRK scheme. Int. J. Mech. Control 21, 163–175 (2020)
Jena, S.R., Senapati, A., Gebremedhin, G.S.: Approximate solution of MRLW equation in B-spline environment. Math. Sci. 14, 345–357 (2020)
Lavrentiev, M.M., Romanov, V.G., Vasiliev, V.G.: Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (1970)
Mathworks: Documentation optimization toolbox-least squares algorithms, 2019. Available at www.mathworks.com
Megraliev, Y.T., Alizade, F.K.: Inverse boundary value problem for a Boussinesq type equation of fourth order with nonlocal time integral conditions of the second kind. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauk. 26, 503–514 (2016)
Mittal, R.C., Arora, G.: Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation. Commun. Nonlinear Sci. Numer. Simul. 15, 2798–2808 (2010)
Mittal, R.C., Jain, R.K.: B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations. Appl. Math. Comput. 217, 9741–9755 (2011)
Mohanty, M., Jena, S.R.: Differential transformation method for approximate solution of ordinary differential equation. Adv. Model. Anal. B 61, 135–138 (2018)
Mohanty, M., Jena, S.R., Mishra, S.K.: Mathematical modelling in engineering with integral transforms via modified Adomian decomposition method. Math. Model. Eng. Probl. 8, 409–417 (2021)
Mohanty, M., Jena, S.R., Mishra, S.K.: Approximate solution of fourth order differential equation. Adv. Math. 10, 621–628 (2021)
O’Brien, G.G., Hyman, M.A., Kaplan, S.: A study of the numerical solution of partial differential equations. J. Math. Phys. 29, 223–251 (1950)
Rodriguez, P.: Total variation regularization algorithms for images corrupted with different noise models: a review. J. Electr. Comput. Eng. 2013, Article ID 217021, 18 pages (2013)
Senapati, A., Jena, S.R.: A computational scheme for fifth order boundary value problems. Int. J. Inf. Technol. 14, 1397–1404 (2022). https://doi.org/10.1007/s41870-022-00871-7
Tikhonov, A.N.: On the stability of inverse problems. Dokl. Akad. Nauk SSSR 39, 195–198 (1943)
Vichnevetsky, R.: Stability charts in the numerical approximation of partial differential equations: a review. Math. Comput. Simul. 21, 170–177 (1979)
Wang, Y., Yang, C., Yagola, A.: Optimization and Regularization for Computational Inverse Problems and Applications. Springer, Berlin (2011)
Yang, H.: An inverse problem for the sixth-order linear Boussinesq-type equation. UPB Sci. Bull., Ser. A 82, 27–36 (2020)