Document Type : Research Paper


1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

2 Department of Mathematics, Islamic Azad University, Lahijan Branch, Lahijan, Iran.


In this article, a mathematical model of the inverse problem is considered. Based on this model a formulation of inverse problem for heat equation is proposed. Shifted Chebyshev Tau (SCT) method is suggested to solve the inverse problem. The aim of this determined effort is to identify unknown function and unknown control parameter of the mathematical model. In order to achieve highly accurate solution to this problem, the operational matrix of shifted Chebyshev polynomials is investigated in conjunction with tau scheme. To demonstrate the validity and applicability of the developed scheme, numerical example is presented.


Main Subjects

[1]     Cannon, J. R., Lin, Y., & Wang, S. (1991). Determination of a control parameter in a parabolic partial differential equation. The anzim journal, 33(2), 149–163. DOI:10.1017/S0334270000006962
[2]     Akbarpour, S., Shidfar, A., & Saberinajafi, H. (2020). A Shifted Chebyshev-Tau method for finding a time-dependent heat source in heat equation. Computational methods for differential equations, 8(1), 1–13.
[3]     Saadatmandi, A., Dehghan, M., & Campo, A. (2006). The Legendre-tau technique for the determination of a source parameter in a semilinear parabolic equation. Mathematical problems in engineering, 2006.
[4]     Dehghan, M., Shafieeabyaneh, N., & Abbaszadeh, M. (2021). A local meshless procedure to determine the unknown control parameter in the multi-dimensional inverse problems. Inverse problems in science and engineering, 29(10), 1369–1400. DOI:10.1080/17415977.2020.1849180
[5]     Mohebbi, A., & Dehghan, M. (2010). High-order scheme for determination of a control parameter in an inverse problem from the over-specified data. Computer physics communications, 181(12), 1947–1954.
[6]     Shidfar, A., Zolfaghari, R., & Damirchi, J. (2009). Application of sinc-collocation method for solving an inverse problem. Journal of computational and applied mathematics, 233(2), 545–554.
[7]     Tatari, M., & Dehghan, M. (2007). Identifying a control function in parabolic partial differential equations from overspecified boundary data. Computers & mathematics with applications, 53(12), 1933–1942.
[8]     Hazanee, A., Lesnic, D., Ismailov, M. I., & Kerimov, N. B. (2019). Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions. Applied mathematics and computation, 346, 800–815.
[9]     Ewing, R. E., & Lin, T. (1991). A class of parameter estimation techniques for fluid flow in porous media. Advances in water resources, 14(2), 89–97.
[10]   Hohage, T. (2006). Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem. Journal of computational physics, 214(1), 224–238.
[11]   Hussein, E. M. (2019). Effect of fractional parameter on thermoelastic half-space subjected to a moving heat source. International journal of heat and mass transfer, 141, 855–860.
[12]   Ismailov, M. I., & Çiçek, M. (2016). Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions. Applied mathematical modelling, 40(7-8), 4891–4899.
[13]   Kolodziej, J. A., Jankowska, M. A., & Mierzwiczak, M. (2013). Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment. International journal of solids and structures, 50(25-26), 4217–4225.
[14]   Li, F. L., Wu, Z. K., & Ye, C. R. (2012). A finite difference solution to a two-dimensional parabolic inverse problem. Applied mathematical modelling, 36(5), 2303–2313.
[15]   Xiao, C., Liu, J., & Liu, Y. (2011). An inverse pollution problem in porous media. Applied mathematics and computation, 218(7), 3649–3653.
[16]   Hazanee, A., Lesnic, D., Ismailov, M. I., & Kerimov, N. B. (2015). An inverse time-dependent source problem for the heat equation with a non-classical boundary condition. Applied mathematical modelling, 39(20), 6258–6272.
[17]   Zolfaghari, R., & Shidfar, A. (2015). Restoration of the heat transfer coefficient from boundary measurements using the Sinc method. Computational and applied mathematics, 34(1), 29–44.
[18]   Zolfaghari, R. (2013). Parameter determination in a parabolic inverse problem in general dimensions. Computational methods for differential equations, 1(1), 55–70.
[19]   Xiong, Z., Deng, K., Liu, Z., Liu, Y., & Yan, X. (2015). The finite volume element method for a parameter identification problem. Journal of ambient intelligence and humanized computing, 6(5), 533–539.
[20]   Khan, M. N., Siraj-ul-Islam, Hussain, I., Ahmad, I., & Ahmad, H. (2021). A local meshless method for the numerical solution of space-dependent inverse heat problems. Mathematical methods in the applied sciences, 44(4), 3066–3079.
[21]   Ashpazzadeh, E., Lakestani, M., & Razzaghi, M. (2017). Cardinal Hermite interpolant multiscaling functions for solving a parabolic inverse problem. Turkish journal of mathematics, 41(4), 1009–1026.
[22]   Grimmonprez, M., Marin, L., & Van Bockstal, K. (2020). The reconstruction of a solely time-dependent load in a simply supported non-homogeneous Euler–Bernoulli beam. Applied mathematical modelling, 79, 914–933.
[23]   Siraj-ul-Islam, & Ismail, S. (2017). Meshless collocation procedures for time-dependent inverse heat problems. International journal of heat and mass transfer, 113, 1152–1167.
[24]   Lakestani, M., & Dehghan, M. (2010). The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement. Journal of computational and applied mathematics, 235(3), 669–678.
[25]   Liao, W., Dehghan, M., & Mohebbi, A. (2009). Direct numerical method for an inverse problem of a parabolic partial differential equation. Journal of computational and applied mathematics, 232(2), 351–360.
[26]   Saadatmandi, A., & Dehghan, M. (2012). A method based on the tau approach for the identification of a time-dependent coefficient in the heat equation subject to an extra measurement. Journal of vibration and control, 18(8), 1125–1132.
[27]   Yang, X., Jiang, X., & Zhang, H. (2018). A time–space spectral tau method for the time fractional cable equation and its inverse problem. Applied numerical mathematics, 130, 95–111.
[28]   Bhrawy, A. H., & Alofi, A. S. (2013). The operational matrix of fractional integration for shifted Chebyshev polynomials. Applied mathematics letters, 26(1), 25–31.
[29]   Atabakzadeh, M. H., Akrami, M. H., & Erjaee, G. H. (2013). Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations. Applied mathematical modelling, 37(20-21), 8903–8911.
[30]   Doha, E., Bhrawy, A., & Ezz-Eldien, S. (2013). Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method. Open physics, 11(10), 1494–1503.
[31]   Bhrawy, A. H., & Zaky, M. A. (2015). A method based on the Jacobi tau approximation for solving multi-term time–space fractional partial differential equations. Journal of computational physics, 281, 876–895.
[32]   Davoodi, F., Abbas Nejad, A., Shahrezaee, A., & Maghrebi, M. J. (2011). Control parameter estimation in a semi-linear parabolic inverse problem using a high accurate method. Applied mathematics and computation, 218(5), 1798–1804.