scholarly journals On Effects of a New Method for Fractional Initial Value Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Hülya Kodal Sevindir ◽  
Süleyman Çetinkaya ◽  
Ali Demir
Keyword(s):  
Iterative Method ◽  
Fractional Derivative ◽  
Infinite Series ◽  
Initial Value Problems ◽  
Analytic Solution ◽  
Initial Value ◽  
Transform Method ◽  
Series Form ◽  
Time Fractional Derivative ◽  
A New Technique

The aim of this study is to analyze nonlinear Liouville-Caputo time-fractional problems by a new technique which is a combination of the iterative and ARA transform methods and is denoted by IAM. First, the ARA transform method and its inverse are utilized to get rid of time fractional derivative. Later, the iterative method is applied to establish the solution of the problem in infinite series form. The main advantages of this method are that it converges to analytic solution of the problem rapidly and implementation of method is easy. Finally, outcomes of the illustrative examples prove the efficiency and accuracy of the method.

A Quasi Trefftz-Type Spectral Method for Initial Value Problems with Moving Boundaries

1997 ◽  
Vol 07 (03) ◽  
pp. 385-404 ◽  
Author(s):  
S. A. Lifits ◽  
S. Yu. Reutskiy ◽  
B. Tirozzi
Keyword(s):  
Spectral Method ◽  
Initial Value Problems ◽  
Analytic Solution ◽  
Complex Geometry ◽  
High Accuracy ◽  
Moving Boundaries ◽  
Initial Value ◽  
Stefan Problems ◽  
A New Technique ◽  
Boundary Methods

A quasi Trefftz-type spectral method is a new technique for numerical solving boundary value and initial value problems in domains with complex geometry. QTSM combines the properties of the boundary methods with the spectral approach. In the present paper QTSM is applied to the problems with moving boundaries. The problems with a known boundary motion law as well as the Stefan problems are considered. The method is tested on several one- and two-dimensional problems with exact analytic solution. A high accuracy of the calculations is achieved.

scholarly journals Approximate solution of nonlinear ordinary differential equation using ZZ decomposition method

2020 ◽  
Vol 4 (1) ◽  
pp. 448-455
Author(s):  
Mulugeta Andualem ◽  
◽  
Atinafu Asfaw ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Approximate Solution ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Initial Value ◽  
Transform Method ◽  
Adomian Decomposition

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

scholarly journals TheZ-Transform Method and Delta Type Fractional Difference Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Matrix Function ◽  
Initial Value Problems ◽  
Stability Problem ◽  
Difference Operators ◽  
Fractional Order Systems ◽  
Initial Value ◽  
Fractional Difference ◽  
Transform Method ◽  
Delta Type ◽  
The Stability

The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classicalZ-transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in theZ-transform. We also prove forms of images in theZ-transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.

scholarly journals Relationship between Sumudu and Some Efficient Integral Transforms

2020 ◽  
Vol 9 (3) ◽  
pp. 153-159 ◽  
Keyword(s):  
Infinite Series ◽  
Initial Value Problems ◽  
Integral Transforms ◽  
Renewal Equation ◽  
Physical Explanation ◽  
Initial Value ◽  
Gamma Functions ◽  
Improper Integrals ◽  
The Relationship

Nowadays integral transforms are most appropriate techniques for finding the solution of typical problems because these techniques convert them into simpler problems. Finding the solution of initial value problems is the main use of integral transforms. However, there are so many other applications of integral transforms in different areas of mathematics and statistics such as in solving improper integrals of first kind, evaluating the sum of the infinite series, developing the relationship between Beta and Gamma functions, solving renewal equation etc. In this paper, scholars established the relationship between Sumudu and some efficient integral transforms. The application section of this paper has tabular representation of integral transforms of some regularly used functions to demonstrate the physical explanation of relationship between Sumudu and mention integral transforms.

Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function

2021 ◽  
Vol 24 (4) ◽  
pp. 1220-1230
Author(s):  
Mohammed Al-Refai
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Maximum Principles ◽  
Initial Value ◽  
Derivative Operator ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Two Parameters

Abstract In this paper, we formulate and prove two maximum principles to nonlinear fractional differential equations. We consider a fractional derivative operator with Mittag-Leffler function of two parameters in the kernel. These maximum principles are used to establish a pre-norm estimate of solutions, and to derive certain uniqueness and positivity results to related linear and nonlinear fractional initial value problems.

scholarly journals An Iterative Method for Time-Fractional Swift-Hohenberg Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Wenjin Li ◽  
Yanni Pang
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Cauchy Problems ◽  
Initial Value ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

We study a type of iterative method and apply it to time-fractional Swift-Hohenberg equation with initial value. Using this iterative method, we obtain the approximate analytic solutions with numerical figures to initial value problems, which indicates that such iterative method is effective and simple in constructing approximate solutions to Cauchy problems of time-fractional differential equations.

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