## Abstract

We consider an inverse source two-parameter sub-diffusion model subject to a non-local initial condition. The problem models several physical processes, among them are the microwave heating and light propagation in photoelectric cells. A bi-orthogonal pair of bases is employed to construct a series representation of the solution and a Volterra integral equation for the source term. We develop a stable numerical algorithm, based on discontinuous collocation method, for approximating the unknown time-dependent source term. Due to the singularity of the solution near \(t=0\), a graded mesh is used to maintain optimal convergence rates, both theoretically and numerically. Numerical experiments are provided to illustrate the expected analytical order of convergence.

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## References

Adams EE, Gelhar LW (1992) Field study of dispersion in heterogeneous aquifer 2. Water Resources Res 28:293–307

Aleroev TS, Kirane M, Malik SA (2013) Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition. Electron J Differ Equ No. 270:1–16

Amir SZ, Sun S (2018) Physics-preserving averaging scheme based on Grünwald-Letnikov formula for gas flow in fractured media. J Petroleum Sci Eng 163:616–639

Baleanu D, Güvenç ZB, Machado JT (eds) (2010) New Trends in Nanotechnology and Fractional Calculus Applications. Springer

Brunner H (2004) Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge

Brunner H, Pedas A, Vainikko G (1999) The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations. Math Comput 68:1079–1096

Cannon JR, Pérez Esteva S, van der Hoek J (1987) A Galerkin procedure for the diffusion equation subject to the specification of mass. SIAM J Nume Anal 24(3):499–515

Cao Y, Herdman T, Xu Y (2003) A hybrid collocation method for volterra integral equations with weakly singular kernels. SIAM J Numer Anal 41:364–381

Caponetto R, Dongola G, Fortuna L, Petráš I (2010) Fractional Order Systems: Modeling and Control Applications, volume 72 of World Scientific Series on Nonlinear Science. World Scientific

Chen X, Chen YM (1997) Efficient algorithm for solving inverse source problems of a nonlinear diffusion equation in microwave heating. J Comput Phys 132:374–383

Cusimano N, Gerardo-Giorda L (2018) A space-fractional Monodomain model for cardiac electrophysiology combining anisotropy and heterogeneity on realistic geometries. J Comput Phys 362:409–424

Demir A, Kanca F (2015) Ozbilge E (2015) Numerical solution and distinguishability in time fractional parabolic equation. Boundary Value Problems 142

Gumel AB (1999) On the numerical solution of the diffusion equation subject to the specification of mass. Aust Mathe Soc J Ser B Appl Math 40(4):475–483

Hazanee A, Lesnic D, Ismailov M, Kerimov NB (2019) Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions. Appl Mathe Comput 346:800–815

Hilfer R editor (2000) Applications of Fractional Calculus in Physics, Singapore, World Scientific

Hilfer R (2000) Fractional diffusion based on Riemann-Liouville fractional derivatives. J Phys Chem B 104(16):3914–3917

Hilfer R Fractional time evolution. In Applications of Fractional Calculus in Physics [15], pages 87–130

Hu X, Zhao L, Shaikh AW (2007) The boundary penalty method for the diffusion equation subject to the specification of mass. Appl Math Comput 186(1):735–748

Ismailov MI, Çiçek M (2016) Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions. Appl Math Modelling 40:4891–4899

Jin B, Rundell W (2015) A tutorial on inverse problems for anomalous diffusion processes. Inverse Problems 31(3):035003

Kamocki R (2016) A new representation formula for the Hilfer fractional derivative and its application. J Comput Appl Math 308:39–45

Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and Applications of Fractional Differential Equations, volume 204 of Mathematics Studies. Elsevier, Amsterdam

Klages R, Radons G, Sokolov I (2008) editors. Anomalous Transport: Foundations and Applications. Wiley

Kleefeld A, Vorderwülbecke S, Burgeth B (2018) Anomalous diffusion, dilation, and erosion in image processing. Int J Comput Math 95(6–7):1375–1393

Linz P (1985) Analytical and numerical methods for Volterra equations. SIAM Studies in Applied Mathematics, SIAM, Philadelphia

Lubich C (2004) Convolution quadrature revisited. BIT Nume Math 44:503–514

Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London

Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77

Michelitsch TM, Collet BA, Riascos AP, Nowakowski AF, Nicolleau FCGA (2017) Fractional random walk lattice dynamics. J Phys A 50(5):055003

Monje CA, Chen Y, Vinagre BM, Xue D, Feliu V (2010) Fractional-order systems and controls. advances in industrial control. Springer, Berlin

Mustapha K (2013) A superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels. Math Comput 82:1987–2005

Mustapha K, Ryan JK (2013) Post-processing discontinuous Galerkin solutions to Volterra integro-differential equations: Analysis and simulations. J Comput Appl Math 253:89–103

Ortigueira MD (2011) Fractional Calculus for Scientists and Engineers, vol 84. Lecture Notes in Electrical Engineering. Springer

Özkum G, Demir A, Erman S, Korkmaz E, Özgür B (2013) On the inverse problem of the fractional heat-like partial differential equations: determination of the source function. Advances in Mathematical Physics, Article ID 476154:8 pages

Petrás̆ (2011) Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer

Podlubny I (1999) Fractional Differential Equations, volume 198 of Mathematics in Science and Engineering. Academic Press

Prabhakar TR (1971) A singular integral equation with a generalized mittag-leffler function in the kernel. Yokohama Math J 19:7–15

Sakamoto K, Yamamoto M (2011) Inverse source problem with a final overdetermination for a fractional diffusion equation. Math Control Related Fields 1(4):509–518

Wang N, Zhou H, Chen H, Xia M, Wang S, Fang J, Sun P (2018) A constant fractional-order viscoelastic wave equation and its numerical simulation scheme. Geophysics 83(1):T39–T48

Wei T, Zhang ZQ (2013) Reconstruction of a time-dependent source term in a time-fractional diffusion equation. Eng Anal Boundary Elements 37(1):23–31

Yang F, Fu C-L, Li X-X (2015) The inverse source problem for time-fractional diffusion equation: stability analysis and regularization. Inverse Prob Sci Eng 23(6):969–996

Zhou L, Selim HM (2003) Application of the fractional advection-dispersion equation in porous media. Soil Sci Soc Am J 67(4):1079–1084

## Acknowledgements

The authors would like to acknowledge the support provided by the Deanship of Scientific Research at King Fahd University of Petroleum and Minerals via the Project FT151003.

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Communicated by Roberto Garrappa.

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Furati, K.M., Mustapha, K., Sarumi, I.O. *et al.* Inverse source in two-parameter anomalous diffusion, numerical algorithms, and simulations over graded time meshes.
*Comp. Appl. Math.* **40, **25 (2021). https://doi.org/10.1007/s40314-020-01399-x

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### Keywords

- Microwave heating
- Inverse source problem
- Anomalous diffusion
- Volterra integral equation
- Collocation method
- Graded mesh error analysis

### Mathematics Subject Classification

- 35R11
- 35R30