Two stable methods with numerical experiments for solving the backward heat equation

2011 ◽  
Vol 61 (2) ◽  
pp. 266-284 ◽  
Author(s):  
Fabien Ternat ◽  
Oscar Orellana ◽  
Prabir Daripa
Keyword(s):  
Heat Equation ◽  
Numerical Experiments

scholarly journals Stable Numerical Evaluation of Finite Hankel Transforms and Their Application

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Manoj P. Tripathi ◽  
B. P. Singh ◽  
Om P. Singh
Keyword(s):  
Stability Analysis ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Numerical Evaluation ◽  
Radiation Condition ◽  
Hankel Transform ◽  
Basis Functions ◽  
Infinite Cylinder ◽  
Stable Algorithm ◽  
Hankel Transforms

A new stable algorithm, based on hat functions for numerical evaluation of Hankel transform of order ν>-1, is proposed in this paper. The hat basis functions are used as a basis to expand a part of the integrand, rf(r), appearing in the Hankel transform integral. This leads to a very simple, efficient, and stable algorithm for the numerical evaluation of Hankel transform. The novelty of our paper is that we give error and stability analysis of the algorithm and corroborate our theoretical findings by various numerical experiments. Finally, an application of the proposed algorithm is given for solving the heat equation in an infinite cylinder with a radiation condition.

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale ◽  
Serife Bayramoglu
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Numerical Experiments ◽  
Time Derivative ◽  
Heat Equations ◽  
First Order ◽  
Fractional Heat Equation

AbstractIn this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2-α + h 2), conditionally. Numerical experiments are carried out to support the theoretical claims.

scholarly journals Recovery of dipolar sources and stability estimates

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4095-4114
Author(s):  
Ridha Mdimagh
Keyword(s):  
Heat Flux ◽  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Time Dependent ◽  
Stability Estimates ◽  
Lateral Boundary ◽  
Lipschitz Stability ◽  
Spatial Locations ◽  
Stability Results

The inverse problem of identifying dipolar sources with time-dependent moments, located in a bounded domain, via the heat equation is investigated, by applying a heat flux, and from a single lateral boundary measurement of temperature. An uniqueness, and local Lipschitz stability results for this inverse problem are established which are the main contributions of this work. A non-iterative algebraic algorithm based on the reciprocity gap concept is proposed, which permits to determine the number, the spatial locations, and the time-dependent moments of the dipolar sources, Some numerical experiments are given in order to test the efficiency and the robustness of this method.

scholarly journals Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jone Apraiz ◽  
Jin Cheng ◽  
Anna Doubova ◽  
Enrique Fernández-Cara ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Inverse Problems ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Spatial Dimension ◽  
Numerical Reconstruction ◽  
Spatial Interval ◽  
Boundary Information ◽  
Size Estimates

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

An Adaptive Scheme to Handle the Phenomenon of Quenching for a Localized Semilinear Heat Equation with Neumann Boundary Conditions

2009 ◽  
Vol 9 (4) ◽  
pp. 339-353
Author(s):  
TH. K. Kouakou ◽  
TH. K. Boni ◽  
R. K. Kouakou
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Finite Time ◽  
Numerical Experiments ◽  
Mesh Size ◽  
Neumann Boundary ◽  
Discrete Form ◽  
Adaptive Scheme ◽  
Semilinear Heat Equation ◽  
Quenching Time

AbstractUnder some assumptions, we prove that the solution of a discrete form of the above problem quenches in a finite time and estimate its numerical quenching time. We also show that the numerical quenching time in certain cases converges to the real one when the mesh size tends to zero. Finally, we give some numerical experiments to illustrate our analysis.

scholarly journals Numerical Solution of Parabolic Problems Based on a Weak Space-Time Formulation

2017 ◽  
Vol 17 (1) ◽  
pp. 65-84 ◽  
Author(s):  
Stig Larsson ◽  
Matteo Molteni
Keyword(s):  
Heat Equation ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
A Priori ◽  
Space Time ◽  
Parabolic Problems ◽  
Weak Space ◽  
Time Formulation ◽  
The Difference

AbstractWe investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the ${L^{2}}$ sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings.

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