scholarly journals Using Difference Scheme Method to the Fractional Telegraph Model with Atangana-Baleanu-Caputo Derivative

2019 ◽  
Author(s):  
Mahmut Modanli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Difference Scheme ◽  
Caputo Derivative ◽  
Difference Schemes ◽  
Laplace Method ◽  
Stability Of Difference Schemes ◽  
The Matrix ◽  
Finite Method

The fractional telegraph partial differential equation with fractional Atangana-Baleanu-Caputo (ABC) derivative is studied. Laplace method is used to find the exact solution of this equation. Stability inequalities are proved for the exact solution. Difference schemes for the implicit finite method are constructed. The implicit finite method is used to deal with modelling the fractional telegraph differential equation defined by Caputo fractional of Atangana-Baleanu (AB) derivative for different interval. Stability of difference schemes for this problem is proved by the matrix method. Numerical results with respect to the exact solution confirm the accuracy and effectiveness of the proposed method.

On Solutions of Fractional Telegraph Model with Mittag-Leffler Kernel

2021 ◽  
Author(s):  
Ali Akgül ◽  
Mahmut Modanli
Keyword(s):  
Numerical Experiments ◽  
Telegraph Equation ◽  
Difference Schemes ◽  
Laplace Method ◽  
Stability Of Difference Schemes ◽  
Fractional Telegraph Equation ◽  
The Matrix ◽  
The Difference ◽  
The Stability ◽  
Finite Method

Abstract In this paper, we research the fractional telegraph equation with the Atangana-Baleanu-Caputo derivative. We use the Laplace method to find the exact solution of the problems. We construct the difference schemes for the implicit finite method. We prove the stability of difference schemes for the problems by the matrix method. We demonstrate the accuracy of the method by some numerical experiments. The obtained results confirm the accuracy and effectiveness of the proposed method. Additionally, the numerical results demonstrate that the expected physical properties of the model are also observed.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

A family of positivity preserving schemes for numerical solution of Black–Scholes equation

2016 ◽  
Vol 03 (04) ◽  
pp. 1650025
Author(s):  
M. Mehdizadeh Khalsaraei ◽  
R. Shokri Jahandizi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Numerical Solution ◽  
Numerical Scheme ◽  
Initial Vector ◽  
Positivity Preserving ◽  
Fully Implicit ◽  
Black Scholes ◽  
Spurious Oscillations

When one solves the Black–Scholes partial differential equation, it is of great important that numerical scheme to be free of spurious oscillations and satisfy the positivity requirement. With positivity, we mean, the component non-negativity of the initial vector, is preserved in time for the exact solution. Numerically, such property for fully implicit scheme is not always satisfied by approximated solutions and they generate spurious oscillations in the presence of discontinuous payoff. In this paper, by using the nonstandard discretization strategy, we propose a new scheme that is free of spurious oscillations and satisfies the positivity requirement.

scholarly journals A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 349
Author(s):  
Joël Chaskalovic
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Finite Element ◽  
Finite Elements ◽  
A Priori Estimates ◽  
Probabilistic Approach ◽  
Approximation Error ◽  
Random Variables ◽  
Partial Differential

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation uh(k) performed by Pk Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution u relative to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation uh(k). We, thus, fill this knowledge gap by considering the exact solution u together with its corresponding approximation uh(k) as random variables. By a method of consequence, any function where u and uh(k) are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on Wm,p Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation uh(k) in order to consider their respective Wm,p-norm as a random variable, as well as the Wm,p approximation error with regards to Pk finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements Pk1 and Pk2,(k1<k2).

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