On the inverse problems for a family of integro-differential equations
Abstract
An integro-differential equation involving arbitrary kernel in time variable with a family of non-local boundary condition has been considered. Two inverse source problems for integro-differential equations are formulated and the unique-existence results for the solution of inverse source problems are presented. Some particular examples in support of our analysis are discussed.
Keyword : inverse problems, generalized diffusion equation, Bi-orthogonal system of functions, multinomial Mittag-Leffler type functions
How to Cite
Suhaib, K., Ilyas, A., & Malik, S. A. (2023). On the inverse problems for a family of integro-differential equations. Mathematical Modelling and Analysis, 28(2), 255–270. https://doi.org/10.3846/mma.2023.16139
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
E. Bazhlekova and I. Bazhlekov. Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation. J. Comput. Appl. Math., 386:113213, 2021. https://doi.org/10.1016/j.cam.2020.113213
A.V. Chechkin, R. Gorenflo and I.M. Sokolov. Fractional diffusion in inhomogeneous media. J. Phys. A Math. Theor., 38(42):679–684, 2005. https://doi.org/10.1088/0305-4470/38/42/l03
B.D. Coleman and M.E. Gurtin. Equipresence and constitutive equations for rigid heat conductors. Z. fur Angew. Math. Phys., 18(19):199–208, 1967. https://doi.org/10.1007/bf01596912
A. Favini, G.R. Goldstein and J.A. Goldstein. The heat equation with generalized Wentzell boundary condition. J. Evol. Equ., 2(1):1–19, 2002. https://doi.org/10.1007/s00028-002-8077-y
S. Guerrero and O.Y. Imanuvilov. Remarks on non controllability of the heat equation with memory. ESAIM Control Optim. Calc. Var., 19(1):288–300, 2013. https://doi.org/10.1051/cocv/2012013
G.J. Habetler and R.L. Schiffman. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Computing, 6(3):342–348, 1970.
T. Hintermann. Evolution equations with dynamic boundary conditions. Proc. R. Soc. Edinb. A: Math., 113(1-2):43–60, 1989. https://doi.org/10.1017/s0308210500023945
A. Ilyas, S.A. Malik and S. Saif. Inverse problems for a multi-term time fractional evolution equation with an involution. Inverse Probl. Sci. Eng., 29(13):3377– 3405, 2021. https://doi.org/10.1080/17415977.2021.2000606
M.I. Ismailov and F. Kanca. An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions. Math. Methods Appl. Sci., 34(6):692–702, 2011. https://doi.org/10.1002/mma.1396
M.I. Ismailov, I. Tekin and S. Erkovan. An inverse coefficient problem of finding the lowest term for heat equation with Wentzell-Neumann boundary conditions. Inverse Probl. Sci. Eng., 27(11):1608–1634, 2019. https://doi.org/10.1080/17415977.2018.1553968
N.B. Kerimov and M.I. Ismailov. Direct and inverse problems for the heat equation with a dynamic-type boundary condition. IMA J. Appl. Math., 80(5):1519– 1533, 2015. https://doi.org/10.1093/imamat/hxv005
N. Kinash and J. Janno. Inverse problems for a generalized subdiffusion equation with final overdetermination. Math. Model. Anal., 24(2):236–262, 2019. https://doi.org/10.3846/mma.2019.016
S. Larsson, V. Thomee and L.B. Wahlbin. Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math. Comput., 67(221):45–71, 1998. https://doi.org/10.1090/S0025-5718-98-00883-7
X. Li, Q. Xu and A. Zhu. Weak galerkin mixed finite element methods for parabolic equations with memory. Discrete Contin. Dyn. Syst. - S, 12(3):513– 531, 2019. https://doi.org/10.3934/dcdss.2019034
K. Liao and T. Wei. Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously. Inverse Probl., 35(11):115002, 2019. https://doi.org/10.1088/1361-6420/ab383f
T.N. Luana and T.Q. Khanh. On the backward problem for parabolic equations with memory. Appl. Anal., 100(7):1414–1431, 2021. https://doi.org/10.1080/00036811.2019.1643013
Y. Luchko and R. Gorenflo. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam., 24(2):207–233, 1999.
S.A. Malik, A. Ilyas and A. Samreen. Simultaneous determination of a source term and diffusion concentration for a multi-term space-time fractional diffusion equation. Math. Model. Anal., 26(3):411–431, 2021. https://doi.org/10.3846/mma.2021.11911
D.B. Marchenkov. Basis property in lp(0,1) of the system of eigenfunctions corresponding to a problem with a spectral parameter in the boundary condition. Differ. Equ., 42(6):905–908, 2006. https://doi.org/10.1134/S0012266106060152
A.Y. Mokin. On a family of initial-boundary value problems for the heat equation. Differ. Equ., 45(1):126–141, 2009. https://doi.org/10.1134/s0012266109010133
J.W. Nunziato. On heat conduction in materials with memory. Q. Appl. Math., 29(2):187–204, 1971. https://doi.org/10.1090/qam/295683
B.G. Pachpatte. On a nonlinear diffusion system arising in reactor dynamics. J. Math. Anal. Appl., 94(2):501–508, 1983. https://doi.org/10.1016/0022-247X(83)90078-1
M. Slodička. A parabolic inverse source problem with a dynamical boundary condition. Appl. Math. Comput., 256:529–539, 2015. https://doi.org/10.1016/j.amc.2015.01.103
Q. Tao and H. Gao. On the null controllability of heat equation with memory. J. Math. Anal. Appl., 440(1):1–13, 2016. https://doi.org/10.1016/j.jmaa.2016.03.036
H. Wei, W. Chen, H. Sun and X. Li. A coupled method for inverse source problem of spatial fractional anomalous diffusion equations. Inverse Probl. Sci. Eng., 18(7):945–956, 2010. https://doi.org/10.1080/17415977.2010.492515
S. Wei, W. Chen and Y.C. Hon. Characterizing time dependent anomalous diffusion process: A survey on fractional derivative and nonlinear models. Physica A, 462:1244–1251, 2016. https://doi.org/10.1016/j.physa.2016.06.145
E.G. Yanik and G. Fairweather. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal. Theory Methods Appl., 12(8):785–809, 1988. https://doi.org/10.1016/0362-546X(88)90039-9
N.Y. Zhang. On fully discrete Galerkin approximations for partial integrodifferential equations of parabolic type. Math. Comput., 60(201):133–166, 1993. https://doi.org/10.1090/S0025-5718-1993-1149295-4
X. Zhou and H. Gao. Interior approximate and null controllability of the heat equation with memory. Comput. Math. Appl., 67(3):602–613, 2014. https://doi.org/10.1016/j.camwa.2013.12.005
A.V. Chechkin, R. Gorenflo and I.M. Sokolov. Fractional diffusion in inhomogeneous media. J. Phys. A Math. Theor., 38(42):679–684, 2005. https://doi.org/10.1088/0305-4470/38/42/l03
B.D. Coleman and M.E. Gurtin. Equipresence and constitutive equations for rigid heat conductors. Z. fur Angew. Math. Phys., 18(19):199–208, 1967. https://doi.org/10.1007/bf01596912
A. Favini, G.R. Goldstein and J.A. Goldstein. The heat equation with generalized Wentzell boundary condition. J. Evol. Equ., 2(1):1–19, 2002. https://doi.org/10.1007/s00028-002-8077-y
S. Guerrero and O.Y. Imanuvilov. Remarks on non controllability of the heat equation with memory. ESAIM Control Optim. Calc. Var., 19(1):288–300, 2013. https://doi.org/10.1051/cocv/2012013
G.J. Habetler and R.L. Schiffman. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Computing, 6(3):342–348, 1970.
T. Hintermann. Evolution equations with dynamic boundary conditions. Proc. R. Soc. Edinb. A: Math., 113(1-2):43–60, 1989. https://doi.org/10.1017/s0308210500023945
A. Ilyas, S.A. Malik and S. Saif. Inverse problems for a multi-term time fractional evolution equation with an involution. Inverse Probl. Sci. Eng., 29(13):3377– 3405, 2021. https://doi.org/10.1080/17415977.2021.2000606
M.I. Ismailov and F. Kanca. An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions. Math. Methods Appl. Sci., 34(6):692–702, 2011. https://doi.org/10.1002/mma.1396
M.I. Ismailov, I. Tekin and S. Erkovan. An inverse coefficient problem of finding the lowest term for heat equation with Wentzell-Neumann boundary conditions. Inverse Probl. Sci. Eng., 27(11):1608–1634, 2019. https://doi.org/10.1080/17415977.2018.1553968
N.B. Kerimov and M.I. Ismailov. Direct and inverse problems for the heat equation with a dynamic-type boundary condition. IMA J. Appl. Math., 80(5):1519– 1533, 2015. https://doi.org/10.1093/imamat/hxv005
N. Kinash and J. Janno. Inverse problems for a generalized subdiffusion equation with final overdetermination. Math. Model. Anal., 24(2):236–262, 2019. https://doi.org/10.3846/mma.2019.016
S. Larsson, V. Thomee and L.B. Wahlbin. Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math. Comput., 67(221):45–71, 1998. https://doi.org/10.1090/S0025-5718-98-00883-7
X. Li, Q. Xu and A. Zhu. Weak galerkin mixed finite element methods for parabolic equations with memory. Discrete Contin. Dyn. Syst. - S, 12(3):513– 531, 2019. https://doi.org/10.3934/dcdss.2019034
K. Liao and T. Wei. Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously. Inverse Probl., 35(11):115002, 2019. https://doi.org/10.1088/1361-6420/ab383f
T.N. Luana and T.Q. Khanh. On the backward problem for parabolic equations with memory. Appl. Anal., 100(7):1414–1431, 2021. https://doi.org/10.1080/00036811.2019.1643013
Y. Luchko and R. Gorenflo. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam., 24(2):207–233, 1999.
S.A. Malik, A. Ilyas and A. Samreen. Simultaneous determination of a source term and diffusion concentration for a multi-term space-time fractional diffusion equation. Math. Model. Anal., 26(3):411–431, 2021. https://doi.org/10.3846/mma.2021.11911
D.B. Marchenkov. Basis property in lp(0,1) of the system of eigenfunctions corresponding to a problem with a spectral parameter in the boundary condition. Differ. Equ., 42(6):905–908, 2006. https://doi.org/10.1134/S0012266106060152
A.Y. Mokin. On a family of initial-boundary value problems for the heat equation. Differ. Equ., 45(1):126–141, 2009. https://doi.org/10.1134/s0012266109010133
J.W. Nunziato. On heat conduction in materials with memory. Q. Appl. Math., 29(2):187–204, 1971. https://doi.org/10.1090/qam/295683
B.G. Pachpatte. On a nonlinear diffusion system arising in reactor dynamics. J. Math. Anal. Appl., 94(2):501–508, 1983. https://doi.org/10.1016/0022-247X(83)90078-1
M. Slodička. A parabolic inverse source problem with a dynamical boundary condition. Appl. Math. Comput., 256:529–539, 2015. https://doi.org/10.1016/j.amc.2015.01.103
Q. Tao and H. Gao. On the null controllability of heat equation with memory. J. Math. Anal. Appl., 440(1):1–13, 2016. https://doi.org/10.1016/j.jmaa.2016.03.036
H. Wei, W. Chen, H. Sun and X. Li. A coupled method for inverse source problem of spatial fractional anomalous diffusion equations. Inverse Probl. Sci. Eng., 18(7):945–956, 2010. https://doi.org/10.1080/17415977.2010.492515
S. Wei, W. Chen and Y.C. Hon. Characterizing time dependent anomalous diffusion process: A survey on fractional derivative and nonlinear models. Physica A, 462:1244–1251, 2016. https://doi.org/10.1016/j.physa.2016.06.145
E.G. Yanik and G. Fairweather. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal. Theory Methods Appl., 12(8):785–809, 1988. https://doi.org/10.1016/0362-546X(88)90039-9
N.Y. Zhang. On fully discrete Galerkin approximations for partial integrodifferential equations of parabolic type. Math. Comput., 60(201):133–166, 1993. https://doi.org/10.1090/S0025-5718-1993-1149295-4
X. Zhou and H. Gao. Interior approximate and null controllability of the heat equation with memory. Comput. Math. Appl., 67(3):602–613, 2014. https://doi.org/10.1016/j.camwa.2013.12.005