scholarly journals Hysteresis and Phase Transitions in a Lattice Regularization of an Ill-Posed Forward–Backward Diffusion Equation

2018 ◽  
Vol 230 (1) ◽  
pp. 231-275
Author(s):  
Michael Helmers ◽  
Michael Herrmann
Keyword(s):  
Phase Transitions ◽  
Diffusion Equation ◽  
Lattice Regularization ◽  
Ill Posed

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

Quasi solution of a backward space fractional diffusion equation

2019 ◽  
Vol 27 (6) ◽  
pp. 795-814 ◽  
Author(s):  
Amir Hossein Salehi Shayegan ◽  
Ali Zakeri
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Linear Equations ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
The Finite Element Method ◽  
System Of Linear Equations ◽  
Solution Approach ◽  
Space Fractional Diffusion Equation ◽  
Ill Posed

Abstract In this paper, based on a quasi solution approach, i.e., a methodology involving minimization of a least squares cost functional, we study a backward space fractional diffusion equation. To this end, we give existence and uniqueness theorems of a quasi solution in an appropriate class of admissible initial data. In addition, in order to approximate the quasi solution, the finite element method is used. Since the obtained system of linear equations is ill-posed, we apply TSVD regularization. Finally, three numerical examples are given. Numerical results reveal the efficiency and applicability of the proposed method.

scholarly journals Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

2020 ◽  
Vol 28 (2) ◽  
pp. 211-235
Author(s):  
Tran Bao Ngoc ◽  
Nguyen Huy Tuan ◽  
Mokhtar Kirane
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Numerical Example ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e., the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak a priori assumptions on the sought solution, we propose a new regularization method for stabilizing the ill-posed problem. We also provide a numerical example to illustrate our results.

scholarly journals Identifying the space source term problem for time-space-fractional diffusion equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Erdal Karapinar ◽  
Devendra Kumar ◽  
Rathinasamy Sakthivel ◽  
Nguyen Hoang Luc ◽  
N. H. Can
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Choice Rule ◽  
Parameter Choice ◽  
Time Space ◽  
Space Fractional Diffusion Equation ◽  
Ill Posed ◽  
Regularized Solution

Abstract In this paper, we consider an inverse source problem for the time-space-fractional diffusion equation. Here, in the sense of Hadamard, we prove that the problem is severely ill-posed. By applying the quasi-reversibility regularization method, we propose by this method to solve the problem (1.1). After that, we give an error estimate between the sought solution and regularized solution under a prior parameter choice rule and a posterior parameter choice rule, respectively. Finally, we present a numerical example to find that the proposed method works well.

scholarly journals Application of He's Homotopy Perturbation Method for Cauchy Problem of Ill-Posed Nonlinear Diffusion Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Ali Zakeri ◽  
Azim Aminataei ◽  
Qodsiyeh Jannati
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Perturbation Method ◽  
Nonlinear Diffusion ◽  
Homotopy Perturbation Method ◽  
Finite Interval ◽  
Nonlinear Diffusion Equation ◽  
Third Order ◽  
Homotopy Perturbation ◽  
Ill Posed

We consider a Cauchy problem of unidimensional nonlinear diffusion equation on finite interval. This problem is ill-posed and its approximate solution is unstable. We apply the He's homotopy perturbation method (HPM) and obtain the third-order asymptotic expansion. We show that if the conductivity term in diffusion equation has a specified condition, the above solution can be estimated. Finally, a numerical experiment is provided to illustrate the method.

Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity

2021 ◽  
Vol 24 (4) ◽  
pp. 1112-1129
Author(s):  
Dinh Nguyen Duy Hai
Keyword(s):  
Thermal Conductivity ◽  
Diffusion Equation ◽  
Diffusion Problem ◽  
Filter Function ◽  
Convergence Estimate ◽  
Backward Problem ◽  
Special Cases ◽  
Ill Posed ◽  
The Fourier Transform ◽  
Regularized Solution

Abstract This paper concerns a backward problem for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity. Such a problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2), which is usually used to model the anomalous diffusion. We show that the problem is severely ill-posed. Using the Fourier transform and a filter function, we construct a regularized solution from the data given inexactly and explicitly derive the convergence estimate in the case of the local Lipschitz reaction term. Special cases of the regularized solution are also presented. These results extend some earlier works on the space-fractional backward diffusion problem.

scholarly journals IRREVERSIBILITY AND HYSTERESIS FOR A FORWARD–BACKWARD DIFFUSION EQUATION

2004 ◽  
Vol 14 (11) ◽  
pp. 1599-1620 ◽  
Author(s):  
L. C. EVANS ◽  
M. PORTILHEIRO
Keyword(s):  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
Cubic Nonlinearity ◽  
Important Work ◽  
Ill Posed ◽  
Entropy Flux ◽  
Formal Limit

Our intention in this paper is to publicize and extend somewhat important work of Plotnikov7 on the asymptotic limits of solutions of viscous regularizations of a nonlinear diffusion PDE with a cubic nonlinearity. Since the formal limit PDE is in general ill-posed, we expect that the limit solves instead a corresponding diffusion equation with hysteresis effects. We employ entropy/entropy flux pairs to prove various assertions consistent with this expectation.

scholarly journals On an initial inverse problem for a diffusion equation with a conformable derivative

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Tran Thanh Binh ◽  
Nguyen Hoang Luc ◽  
Donal O’Regan ◽  
Nguyen H. Can
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
Conformable Derivative ◽  
Parameter Choice ◽  
Backward Problem ◽  
Ill Posed ◽  
Two Parameter ◽  
Parameter Choice Rules ◽  
Regularized Solution

AbstractIn this paper, we consider the initial inverse problem for a diffusion equation with a conformable derivative in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

scholarly journals Machine-learning based discovery of missing physical processes in radiation belt modeling

2021 ◽  
Author(s):  
Enrico Camporeale ◽  
George Wilkie ◽  
Alexander Drozdov ◽  
Jacob Bortnik
Keyword(s):  
Diffusion Equation ◽  
Radiation Belt ◽  
Radiation Belts ◽  
Artificial Satellites ◽  
Observation Data ◽  
Physical Processes ◽  
Drift Diffusion ◽  
Time Prediction ◽  
Ill Posed ◽  
Physical Insight

Abstract Real-time prediction of the dynamics of energetic electrons in Earth's radiation belts incorporating incomplete observation data is important to protect valuable artificial satellites and to understand their physical processes. Traditionally, reduced models have employed a diffusion equation based on the quasi-linear approximation. Using a Physics-Informed Neutral Network (PINN) framework, we train and test a model based on four years of Van Allen Probe data. We present a recipe for gleaning physical insight from solving the ill-posed inverse problem of inferring model coefficients from data using PINNs. With this, it is discovered that the dynamics of ``killer electrons'' is described more accurately instead by a drift-diffusion equation. A parameterization for the diffusion and drift coefficients, which is both simpler and more accurate than existing models, is presented.

Sign in / Sign up

Export Citation Format

Share Document