On the inverse problems for a family of integro-differential equations

    Kamran Suhaib   Affiliation
    ; Asim Ilyas   Affiliation
    ; Salman A. Malik   Affiliation


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.
Published in Issue
Mar 21, 2023
Abstract Views
PDF Downloads
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.


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.

A.V. Chechkin, R. Gorenflo and I.M. Sokolov. Fractional diffusion in inhomogeneous media. J. Phys. A Math. Theor., 38(42):679–684, 2005.

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.

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.

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.

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.

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.

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.

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.

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.

N. Kinash and J. Janno. Inverse problems for a generalized subdiffusion equation with final overdetermination. Math. Model. Anal., 24(2):236–262, 2019.

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.

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.

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.

T.N. Luana and T.Q. Khanh. On the backward problem for parabolic equations with memory. Appl. Anal., 100(7):1414–1431, 2021.

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.

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.

A.Y. Mokin. On a family of initial-boundary value problems for the heat equation. Differ. Equ., 45(1):126–141, 2009.

J.W. Nunziato. On heat conduction in materials with memory. Q. Appl. Math., 29(2):187–204, 1971.

B.G. Pachpatte. On a nonlinear diffusion system arising in reactor dynamics. J. Math. Anal. Appl., 94(2):501–508, 1983.

M. Slodička. A parabolic inverse source problem with a dynamical boundary condition. Appl. Math. Comput., 256:529–539, 2015.

Q. Tao and H. Gao. On the null controllability of heat equation with memory. J. Math. Anal. Appl., 440(1):1–13, 2016.

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.

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.

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.

N.Y. Zhang. On fully discrete Galerkin approximations for partial integrodifferential equations of parabolic type. Math. Comput., 60(201):133–166, 1993.

X. Zhou and H. Gao. Interior approximate and null controllability of the heat equation with memory. Comput. Math. Appl., 67(3):602–613, 2014.