scholarly journals Using T˜-Transformation for Solving Tank and Heating System Equations

2021 ◽  
Vol 8 (3) ◽  
pp. 441-446
Author(s):  
Rehab A. Khudair ◽  
Ameera N. Alkiffai ◽  
Ahmed S. Sleibi
Keyword(s):  
Differential Equations ◽  
Real Life ◽  
Heating System ◽  
Computational Power ◽  
Fuzzy Transform ◽  
Constant Coefficients ◽  
System Equations ◽  
Fractional Differential ◽  
System Information ◽  
A New Technique

In this article, a fuzzy Tarig evolve (T-n-transform) is implemented. Similar theorems and properties have been proven. To explain the technique of this fuzzy transform in differential equations, examples in real life are presented. This study shows the applicability of this interesting fuzzy transform for solving differential equations with constant coefficients also for its computational power. It is desirable to use it as a new technique, to not only solve “nonlinear fractional differential equations", and to analyze prelocal system information. Moreover, significant theorems are presented to explain the properties of T˜-transform as well as a suggested method is validated with two reality examples.

scholarly journals Solving the Vibrating Spring Equation Using Fuzzy Elzaki Transform

2020 ◽  
Vol 7 (4) ◽  
pp. 549-555
Author(s):  
Rehab Ali Khudair ◽  
Ameera N. Alkiffai ◽  
Athraa Neamah Albukhuttar
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Computational Power ◽  
Fuzzy Transform ◽  
Generalized Differentiability ◽  
Constant Coefficients ◽  
Physical Fields ◽  
New Formula

In this article, a fuzzy Elzaki transform (FZT) is discussed in the context of highly-generalized differentiability concepts, where a new formula of fuzzy derivatives for the fuzzy Elzaki transform is derived as well. It shows the applicability of this interesting fuzzy transform for solving differential equations with constant coefficients also for its computational power. Since ordinary linear equations are mostly used in physical fields, the motion of a mass on a vibrating spring problem is solved by using this kind of fuzzy Elzaki transform.

Cauchy problems for some classes of linear fractional differential equations

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Teodor Atanackovic ◽  
Diana Dolicanin ◽  
Stevan Pilipovic ◽  
Bogoljub Stankovic
Keyword(s):  
Differential Equations ◽  
Cauchy Problems ◽  
Fractional Damping ◽  
Constant Coefficients ◽  
Leading Term ◽  
Fractional Differential ◽  
Linear Differential ◽  
Rational Order ◽  
Derivatives Of ◽  
Linear Fractional Differential Equations

AbstractCauchy problems for a class of linear differential equations with constant coefficients and Riemann-Liouville derivatives of real orders, are analyzed and solved in cases when some of the real orders are irrational numbers and when all real orders appearing in the derivatives are rational numbers. Our analysis is motivated by a forced linear oscillator with fractional damping. We pay special attention to the case when the leading term is an integer order derivative. A new form of solution, in terms of Wright’s function for the case of equations of rational order, is presented. An example is treated in detail.

scholarly journals Conformable Laplace Transform of Fractional Differential Equations

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 55 ◽  
Author(s):  
Fernando Silva ◽  
Davidson Moreira ◽  
Marcelo Moret
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Classical Model ◽  
Transform Method ◽  
Fractional Models ◽  
Constant Coefficients ◽  
Fractional Differential ◽  
Linear Differential ◽  
Fractional Orders

In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, the analytical solution for a class of fractional models associated with the logistic model, the von Foerster model and the Bertalanffy model is presented graphically for various fractional orders. The solution of the corresponding classical model is recovered as a particular case.

scholarly journals Some Higher-Degree Lacunary Fractional Splines in the Approximation of Fractional Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 422
Author(s):  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed ◽  
Juan L. G. Guirao ◽  
Y. S. Hamed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Real Life ◽  
Spline Functions ◽  
Spline Method ◽  
Unique Positive Solution ◽  
Caputo Fractional Differential Equations ◽  
Recent Developments ◽  
Fractional Differential

In this article, we begin by introducing two classes of lacunary fractional spline functions by using the Liouville–Caputo fractional Taylor expansion. We then introduce a new higher-order lacunary fractional spline method. We not only derive the existence and uniqueness of the method, but we also provide the error bounds for approximating the unique positive solution. As applications of our fundamental findings, we offer some Liouville–Caputo fractional differential equations (FDEs) to illustrate the practicability and effectiveness of the proposed method. Several recent developments on the the theory and applications of FDEs in (for example) real-life situations are also indicated.

An inverse problem approach to determine possible memory length of fractional differential equations

2021 ◽  
Vol 24 (6) ◽  
pp. 1919-1936
Author(s):  
Chuan–Yun Gu ◽  
Guo–Cheng Wu ◽  
Babak Shiri
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fundamental Problem ◽  
Real Life ◽  
Memory Length ◽  
Nonlinear Alternative ◽  
Starting Point ◽  
Fractional Differential

Abstract It is a fundamental problem to determine a starting point in fractional differential equations which reveals the memory length in real life modeling. This paper describes it by an inverse problem. Fixed point theorems such as Krasnoselskii’s and Schauder type’s and nonlinear alternative for single–valued mappings are presented. Through existence analysis of the inverse problem, the range of the initial value points and the memory length of fractional differential equations are obtained. Finally, three examples are demonstrated to support the theoretical results and numerical solutions are provided.

Improvements in a method for solving fractional integral equations with some links with fractional differential equations

2018 ◽  
Vol 21 (1) ◽  
pp. 174-189 ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Source Term ◽  
Initial Conditions ◽  
Standard Procedure ◽  
Relaxation Equation ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure. In this procedure, there were two major obstacles that did not allow to obtain a full result. These were the assumptions over the smoothness of the source term and the assumption about the rationality of the orders. So, one of the main topics of this document is to describe a modification of the procedure presented in [4], when the source term is not smooth enough to differentiate the required amount of times. Furthermore, we will also study the fractional integral equations with non-rational orders by a limit process of fractional integral equations with rational orders. Finally, we will connect the previous material with some fractional differential equations with Caputo derivatives described in [7]. For instance, we will deal with the fractional oscillation equation, the fractional relaxation equation and, specially, its particular case of the Basset problem. We also expose how to compute these solutions for the Riemann-Liouville case.

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