Inverse source in two-parameter anomalous diffusion, numerical algorithms, and simulations over graded time meshes

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 $$t=0$$ 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.

[1]  I. Podlubny.  Fractional differential equations , 1998 .

[2]  Khaled M. Furati,et al.  An inverse problem for a generalized fractional diffusion , 2014, Appl. Math. Comput..

[3]  Mansur I. Ismailov,et al.  Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions , 2016 .

[4]  B. Burgeth,et al.  Anomalous diffusion, dilation, and erosion in image processing , 2018, Int. J. Comput. Math..

[5]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[6]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[7]  R. Hilfer.  FRACTIONAL TIME EVOLUTION , 2000 .

[8]  W. Marsden.  I and J , 2012 .

[9]  R. Hilfer.  Fractional Diffusion Based on Riemann-Liouville Fractional Derivatives † , 2000 .

[10]  M. I. Ismailov,et al.  Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions , 2019, Appl. Math. Comput..

[11]  Khaled M. Furati,et al.  An inverse source problem for a two-parameter anomalous diffusion with local time datum , 2016, Comput. Math. Appl..

[12]  Masahiro Yamamoto,et al.  Inverse source problem with a finaloverdetermination for a fractional diffusionequation , 2011 .

[13]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[14]  Fan Yang,et al.  The inverse source problem for time-fractional diffusion equation: stability analysis and regularization , 2015 .

[15]  A. P. Riascos,et al.  Fractional random walk lattice dynamics , 2016, 1608.08762.

[16]  William Rundell,et al.  A tutorial on inverse problems for anomalous diffusion processes , 2015, 1501.00251.

[17]  Fernando A. Oliveira,et al.  Anomalous Diffusion , 2008, Thinking Probabilistically.

[18]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[19]  L. Gelhar,et al.  Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis , 1992 .

[20]  E. Eric Adams,et al.  Field study of dispersion in a heterogeneous aquifer , 1992 .

[21]  Manuel Duarte Ortigueira,et al.  Fractional Calculus for Scientists and Engineers , 2011, Lecture Notes in Electrical Engineering.

[22]  Luca Gerardo-Giorda,et al.  A space-fractional Monodomain model for cardiac electrophysiology combining anisotropy and heterogeneity on realistic geometries , 2018, J. Comput. Phys..

[23]  Abba B. Gumel,et al.  On the numerical solution of the diffusion equation subject to the specification of mass , 1999, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[24]  Ting Wei,et al.  Reconstruction of a time-dependent source term in a time-fractional diffusion equation , 2013 .

[25]  Numerical solution and distinguishability in time fractional parabolic equation , 2015 .

[26]  YangQuan Chen,et al.  Fractional-order Systems and Controls , 2010 .

[27]  S. McKee,et al.  Weakly Singular Discrete Gronwall Inequalities , 1986 .

[28]  Rafal Kamocki,et al.  A new representation formula for the Hilfer fractional derivative and its application , 2016, J. Comput. Appl. Math..

[29]  Y. M. Chen,et al.  Efficient Algorithm for Solving Inverse Source Problems of a Nonlinear Diffusion Equation in Microwave Heating , 1997 .

[30]  Shucheng Wang,et al.  A constant fractional-order viscoelastic wave equation and its numerical simulation scheme , 2018 .

[31]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[32]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[33]  Salman A. Malik,et al.  DETERMINATION OF A SOURCE TERM FOR A TIME FRACTIONAL DIFFUSION EQUATION WITH AN INTEGRAL TYPE OVER-DETERMINING CONDITION , 2013 .

[34]  Hermann Brunner,et al.  The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations , 1999, Math. Comput..

[35]  Xiaozhe Hu,et al.  The boundary penalty method for the diffusion equation subject to the specification of mass , 2007, Appl. Math. Comput..

[36]  Yuesheng Xu,et al.  A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels , 2003, SIAM J. Numer. Anal..

[37]  Kassem Mustapha,et al.  A Superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels , 2013, Math. Comput..

[38]  C. Lubich.  Convolution Quadrature Revisited , 2004 .

[39]  Jennifer K. Ryan,et al.  Post-processing discontinuous Galerkin solutions to Volterra integro-differential equations: Analysis and simulations , 2013, J. Comput. Appl. Math..

[40]  Esra Korkmaz,et al.  On the Inverse Problem of the Fractional Heat-Like Partial Differential Equations: Determination of the Source Function , 2013 .

[41]  T. R. Prabhakar.  A SINGULAR INTEGRAL EQUATION WITH A GENERALIZED MITTAG LEFFLER FUNCTION IN THE KERNEL , 1971 .

[42]  John R. Cannon,et al.  A Galerkin procedure for the diffusion equation subject to the specification of mass , 1987 .

[43]  H. M. Selim,et al.  Application of the Fractional Advection‐Dispersion Equation in Porous Media , 2003 .

[44]  Peter Linz,et al.  Analytical and numerical methods for Volterra equations , 1985, SIAM studies in applied and numerical mathematics.

[45]  Editors , 2003 .

[46]  Sahar Z. Amir,et al.  Physics-preserving averaging scheme based on Grünwald-Letnikov formula for gas flow in fractured media , 2018 .

[47]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .