scholarly journals Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Dumitru Baleanu
Keyword(s):  
Parabolic Equation ◽  
Fractional Calculus ◽  
Numerical Solution ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Implicit Schemes ◽  
The Stability

A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions.

scholarly journals Economical factorized schemes for third-order pseudoparabolic equations

2021 ◽  
pp. 44-57
Author(s):  
Мурат Хамидбиевич Бештоков
Keyword(s):  
General Theory ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Third Order ◽  
The Third ◽  
Stability Of Difference Schemes ◽  
The Stability ◽  
Pseudoparabolic Equations

Изучены экономичные факторизованные схемы для псевдопараболических уравнений третьего порядка. На основе общей теории устойчивости разностных схем доказаны устойчивость и сходимость разностных схем. Economical factorized schemes for pseudo-parabolic equations of the third order are studied. On the basis of the general theory of stability of difference schemes, the stability and convergence of difference schemes are proved.

Incomplete Iterative Implicit Schemes

2020 ◽  
Vol 20 (4) ◽  
pp. 727-737 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Approximate Solution ◽  
Iterative Methods ◽  
Parabolic Equations ◽  
Evolutionary Equation ◽  
Finite Dimensional ◽  
Implicit Schemes ◽  
Multidimensional Problems ◽  
Computational Implementation ◽  
The Stability ◽  
Finite Dimensional Hilbert Space

AbstractIn numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions.

Stability of Finite-difference Schemes for Semilinear Multidimensional Parabolic Equations

2012 ◽  
Vol 12 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Bosko Jovanovic ◽  
Magdalena Lapinska-Chrzczonowicz ◽  
Aleh Matus ◽  
Piotr Matus
Keyword(s):  
Finite Difference ◽  
Parabolic Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Nonlinear Odes ◽  
The Stability ◽  
Initial Boundary Value

Abstract Abstract — We have studied the stability of finite-difference schemes approximating initial-boundary value problem (IBVP) for multidimensional parabolic equations with a nonlinear source of a power type. We have obtained simple sufficient input data conditions, in which the solutions of differential and difference problems are globally bounded for all t. It is shown that if these conditions are not satisfied, then the solution can blow-up (go to infinity) in finite time. The lower bound of the blow-up time has been determined. The stability of the difference solution has been proven. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear ODEs, Bihari-type inequalities and their discrete analogs.

Monotone and Economical Difference Schemes on Nonuniform Grids for a Multidimensional Parabolic Equation with Third Kind

2004 ◽  
Vol 4 (3) ◽  
pp. 350-367
Author(s):  
Piotr Matus ◽  
Grigorii Martsynkevich
Keyword(s):  
Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Approximation ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Nonuniform Grids ◽  
The Third ◽  
Priori Estimates ◽  
The Difference

AbstractMonotone economical difference schemes of the second order of local approximation with respect to space variables on nonuniform grids for the heat con- duction equation with the boundary conditions of the third kind in a p-dimensional parallelepiped are constructed. The a priori estimates of stability and convergence of the difference solution in the norm C are obtained by means of the grid maximum principle.

PROBLEMS ON CONJUGATION OF POLYTYPIC EQUATIONS

2001 ◽  
Vol 6 (1) ◽  
pp. 106-116 ◽  
Author(s):  
V. I. Korzyuk ◽  
S. V. Lemeshevsky
Keyword(s):  
Numerical Solution ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Consistency Conditions

Some conjugation problems of hyperbolic and parabolic equations with different consistency conditions on the interface are considered. Issues concerning one‐valued solvability of these problems are considered. Difference schemes for numerical solution of mentioned conjugation problems are proposed. Estimates of accuracy of algorithms suggested are obtained.

scholarly journals A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 806
Author(s):  
Ali Shokri ◽  
Beny Neta ◽  
Mohammad Mehdizadeh Khalsaraei ◽  
Mohammad Mehdi Rashidi ◽  
Hamid Mohammad-Sedighi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Stability Property ◽  
Stability Region ◽  
Mathieu Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Implicit Schemes ◽  
The Stability

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.

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