Numerical Analysis of Direct and Inverse Problems for a Fractional Parabolic Integro-Differential Equation

A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equat...

Full description

Saved in:
Bibliographic Details
Published in:Fractal and fractional 2023-08, Vol.7 (8), p.601
Main Authors: Koleva, Miglena N., Vulkov, Lubin G.
Format: Article
Language:English
Subjects:
Analysis
Bees
Boundary conditions
Caputo derivative
Conjugate gradient method
Differential equations
Diffusion coefficient
finite difference
Finite difference method
Fréchet derivative
Inequality
integro-differential equation
Inverse problems
least-squares discrepancy functional
Mathematical functions
Numerical analysis
Ordinary differential equations
Parabolic differential equations
parabolic PDE-ODE system
Parameter identification
Partial differential equations
Porous media
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed.
ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract7080601