scholarly journals The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Hasan Bulut ◽  
Haci Mehmet Baskonus ◽  
Fethi Bin Muhammad Belgacem
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Ordinary Differential Equations ◽  
Fractional Derivatives ◽  
General Setting ◽  
Two Dimensional ◽  
Transform Method ◽  
Symbolic Algebra ◽  
Sumudu Transform

We introduce the rudiments of fractional calculus and the consequent applications of the Sumudu transform on fractional derivatives. Once this connection is firmly established in the general setting, we turn to the application of the Sumudu transform method (STM) to some interesting nonhomogeneous fractional ordinary differential equations (FODEs). Finally, we use the solutions to form two-dimensional (2D) graphs, by using the symbolic algebra package Mathematica Program 7.

scholarly journals Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Seyma Tuluce Demiray ◽  
Hasan Bulut ◽  
Fethi Bin Muhammad Belgacem
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Simulations ◽  
Ordinary Differential Equations ◽  
Analytical Solutions ◽  
Order Derivative ◽  
Two Dimensional ◽  
Transform Method ◽  
Sumudu Transform ◽  
Fractional Type

We make use of the so-called Sumudu transform method (STM), a type of ordinary differential equations with both integer and noninteger order derivative. Firstly, we give the properties of STM, and then we directly apply it to fractional type ordinary differential equations, both homogeneous and inhomogeneous ones. We obtain exact solutions of fractional type ordinary differential equations, both homogeneous and inhomogeneous, by using STM. We present some numerical simulations of the obtained solutions and exhibit two-dimensional graphics by means of Mathematica tools. The method used here is highly efficient, powerful, and confidential tool in terms of finding exact solutions.

Numerical solution technique for solving isoperimetric variational problems

2020 ◽  
pp. 2150002
Author(s):  
A. M. S. Mahdy ◽  
E. S. M. Youssef
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Calculus Of Variations ◽  
Variational Problems ◽  
Solution Technique ◽  
Transform Method ◽  
Change Technique ◽  
Main Target ◽  
Sumudu Transform

In this paper, we have a zeal for fulfilling the estimated scientific answers for the calculus of variations by using the Sumudu transform method (STM). The main target is to search the numerical arrangement of ordinary differential equations (ODEs) which emerge from the variational problems where first the fundamental condition for the arrangement of the issue is to fulfill the Euler–Lagrange condition and then solve the equations using STM. The valuable properties of the Sumudu change technique are used to downsize the calculation of the issue to a gathering of straight arithmetical conditions. We introduce four variational problems and discover the numerical solution of those problems using STM and plot the curves of those solutions. These models are picked such that there exist systematic answers for them to offer a reasonable diagram and show the effectiveness of the proposed strategy. Numerical outcomes are registered utilizing Maple programming.

scholarly journals Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications

2021 ◽  
Vol 24 (2) ◽  
pp. 518-540
Author(s):  
Hafiz Muhammad Fahad ◽  
Arran Fernandez
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
General Setting ◽  
Fractional Differentiation ◽  
Fractional Calculus Operators ◽  
Liouville Fractional Derivative ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński’s operational calculus approach is used to obtain exact solutions of fractional differential equations with constant coefficients and with this type of fractional derivatives. These solutions can be expressed in terms of Mittag-Leffler type functions.

scholarly journals Exact Solution of Two-Dimensional Fractional Partial Differential Equations

2020 ◽  
Vol 4 (2) ◽  
pp. 21 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Kamil Jassim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Sumudu Transform ◽  
Fractional Differential

In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations.

scholarly journals On a New Integral Transform and Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
A. Kiliçman ◽  
H. Eltayeb
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Integral Transform ◽  
System Of Differential Equations ◽  
Transform Method ◽  
Control Engineering ◽  
Linear Ordinary Differential Equations ◽  
Sumudu Transform ◽  
Integral Transform Method

Integral transform method is widely used to solve the several differential equations with the initial values or boundary conditions which are represented by integral equations. With this purpose, the Sumudu transform was introduced as a new integral transform by Watugala to solve some ordinary differential equations in control engineering. Later, it was proved that Sumudu transform has very special and useful properties. In this paper we study this interesting integral transform and its efficiency in solving the linear ordinary differential equations with constant and nonconstant coefficients as well as system of differential equations.

Application of He’s Homotopy Perturbation Method to Stiff Systems of Ordinary Differential Equations

2008 ◽  
Vol 63 (1-2) ◽  
pp. 19-23 ◽  
Author(s):  
Mohammad Taghi Darvishi ◽  
Farzad Khani
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
Homotopy Perturbation Method ◽  
Nonlinear Ordinary Differential Equations ◽  
Stiff Systems ◽  
Homotopy Perturbation ◽  
Linear And Nonlinear

We propose He’s homotopy perturbation method (HPM) to solve stiff systems of ordinary differential equations. This method is very simple to be implemented. HPM is employed to compute an approximation or analytical solution of the stiff systems of linear and nonlinear ordinary differential equations.

Properties of Spatial-Rotation Transformations of Fractional Derivatives

2020 ◽  
pp. 198-223
Author(s):  
Ehab Malkawi
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Dimensional Space ◽  
Scalar Fields ◽  
Two Dimensional ◽  
Spatial Rotation ◽  
Essential Step ◽  
Derivatives Of

The transformation properties of the fractional derivatives under spatial rotation in two-dimensional space and for both the Riemann-Liouville and Caputo definitions are investigated and derived in their general form. In particular, the transformation properties of the fractional derivatives acting on scalar fields are studied and discussed. The study of the transformation properties of fractional derivatives is an essential step for the formulation of fractional calculus in multi-dimensional space. The inclusion of fractional calculus in the Lagrangian and Hamiltonian dynamical formulation relies on such transformation. Specific examples on the transformation of the fractional derivatives of scalar fields are discussed.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

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