Matrix Approach to Discretization of Ordinary and Partial Differential Equations of Arbitrary Real Order: The Matlab Toolbox

2009 ◽  
Author(s):  
Igor Podlubny ◽  
Tomas Skovranek ◽  
Blas M. Vinagre Jara
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Matrix Approach ◽  
Matlab Toolbox ◽  
Partial Differential ◽  
The Matrix ◽  
Fractional Differential ◽  
Derivatives Of ◽  
Partial Fractional Differential Equations

The method developed recently by Podlubny et al. (I. Podlubny, Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359–386; I. Podlubny et al., Journal of Computational Physics, vol. 228, no. 8, 1 May 2009, pp. 3137–3153) makes it possible to immediately obtain the discretization of ordinary and partial differential equations by replacing the derivatives with their discrete analogs in the form of triangular strip matrices. This article presents a Matlab toolbox that implements the matrix approach and allows easy and convenient discretization of ordinary and partial differential equations of arbitrary real order. The basic use of the functions implementing the matrix approach to discretization of derivatives of arbitrary real order (so-called fractional derivatives, or fractional-order derivatives), and to solution of ordinary and partial fractional differential equations, is illustrated by examples with explanations.

scholarly journals A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

2020 ◽  
Vol 5 (2) ◽  
pp. 35-48 ◽  
Author(s):  
Kamal Ait Touchent ◽  
Zakia Hammouch ◽  
Toufik Mekkaoui
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Invariant Subspace ◽  
Fractional Derivatives ◽  
Subspace Method ◽  
Partial Differential ◽  
Singular Kernels ◽  
Fractional Differential

AbstractIn this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.

scholarly journals A matrix approach to solving hyperbolic partial differential equations using Bernoulli polynomials

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 993-1000 ◽  
Author(s):  
Bicer Erdem ◽  
Salih Yalcinbas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Analysis ◽  
Matrix Equation ◽  
Bernoulli Polynomials ◽  
Hyperbolic Partial Differential Equations ◽  
Matrix Approach ◽  
Residual Function ◽  
Partial Differential ◽  
The Matrix

The present study considers the solutions of hyperbolic partial differential equations. For this, an approximate method based on Bernoulli polynomials is developed. This method transforms the equation into the matrix equation and the unknown of this equation is a Bernoulli coefficients matrix. To demostrate the validity and applicability of the method, an error analysis developed based on residual function. Also examples are presented to illustrate the accuracy of the method.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

scholarly journals The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012031
Author(s):  
E.A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Differential Operators ◽  
Space Time ◽  
Partial Differential ◽  
Stochastic Problems ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Time Fractional Differential Equations

Abstract The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

Darboux operators for linear first-order multi-component equations in arbitrary dimensions

Open Physics ◽  
2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Axel Schulze-Halberg
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
First Order ◽  
Matrix Coefficients ◽  
The Matrix ◽  
Arbitrary Dimensions ◽  
Weyl Equation

AbstractWe construct Darboux operators for linear, multi-component partial differential equations of first order. The number of variables and the dimension of the matrix coefficients in our equations are arbitrary. The Darboux operator and the transformed equation are worked out explicitly. We present an application of our formalism to the (1+2)-dimensional Weyl equation.

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