Wyniki wyszukiwania - BazTech

1

Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

Rasheed Sarbast Kamal, Modanli Mahmut, Abdulazeez Sadeq Taha

Journal of Applied Mathematics and Computational Mechanics

|

2023

|

Vol. 22, nr 3

33--42

EN

This paper examines a third-order fractional partial differential equation (FPDE) in the Caputo sense. The Theta difference method (TDM) is utilized to investigate the problem, and a first-order difference scheme is developed. Stability estimates are obtained by applying the Von Neumann analysis method. A test problem is presented as an application, and numerical results are obtained using Matlab software. Error estimates, as well as exact and approximate solutions are presented in a data analysis table. The simulation results are shown through error analysis tables and figures.

2

The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach

Maayah Banan, Moussaoui Asma, Bushnaq Samia, Arqub Omar Abu

Demonstratio Mathematica

|

2022

|

Vol. 55, nr 1

963--977

EN

COVID-19, a novel coronavirus disease, is still causing concern all over the world. Recently, researchers have been concentrating their efforts on understanding the complex dynamics of this widespread illness. Mathematics plays a big role in understanding the mechanism of the spread of this disease by modeling it and trying to find approximate solutions. In this study, we implement a new technique for an approximation of the analytic series solution called the multistep Laplace optimized decomposition method for solving fractional nonlinear systems of ordinary differential equations. The proposed method is a combination of the multistep method, the Laplace transform, and the optimized decomposition method. To show the ability and effectiveness of this method, we chose the COVID-19 model to apply the proposed technique to it. To develop the model, the Caputo-type fractional-order derivative is employed. The suggested algorithm efficacy is assessed using the fourth-order Runge-Kutta method, and when compared to it, the results show that the proposed approach has a high level of accuracy. Several representative graphs are displayed and analyzed in two dimensions to show the growth and decay in the model concerning the fractional parameter α values. The central processing unit computational time cost in finding graphical results is utilized and tabulated. From a numerical viewpoint, the archived simulations and results justify that the proposed iterative algorithm is a straightforward and appropriate tool with computational efficiency for several coronavirus disease differential model solutions.

3

Extremal solutions to a coupled system of nonlinear fractional differential equations with ψ–Caputo fractional derivatives

Derbazi Choukri, Baitiche Zidane, Benchohra Mouffak, Graef John R.

Journal of Mathematics and Applications

|

2021

|

Vol. 44

19--34

EN

Using the well-known monotone iterative technique together with the method of upper and lower solutions, the authors investigate the existence of extremal solutions to a class of coupled systems of nonlinear fractional differential equations involving the ψ–Caputo derivative with initial conditions. As applications of this work, two illustrative examples are presented.

4

Analysis of solutions of the 1D fractional Cattaneo heat transfer equation

Siedlecka Urszula, Ciesielski Mariusz

Journal of Applied Mathematics and Computational Mechanics

|

2021

|

Vol. 20, nr 4

87--98

EN

In this paper, a solution of the single-phase lag heat conduction problem is presented. The research concerns the generalized 1D Cattaneo equation in a whole-space domain, where a second order time derivative is replaced by the fractional Caputo derivative. The Fourier-Laplace transform technique is used to determine a solution of the considered problem. The numerical inversion of the Laplace transforms is applied. The effect of the order of the fractional derivative on the temperature distribution is investigated.

5

Mittag-Leffler stability for a Timoshenko problem

Tatar Nasser-eddine

International Journal of Applied Mathematics and Computer Science

|

2021

|

Vol. 31, no. 2

219--232

EN

A Timoshenko system of a fractional order between zero and one is investigated here. Using a fractional version of resolvents, we establish an existence and uniqueness theorem in an appropriate space. Moreover, it is proved that lower order fractional terms (in the rotation component) are capable of stabilizing the system in a Mittag-Leffler fashion. Therefore, they deserve to be called damping terms. This is shown through the introduction of some new functionals and some fractional inequalities, and the establishment of some properties, involving fractional derivatives. In the case of different wave speeds of propagation we obtain convergence to zero.

6

Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with ψ-Caputo fractional derivative

Wahash Hanan A., Abdo Mohammed S, Panchal Satish K.

Journal of Applied Mathematics and Computational Mechanics

|

2020

|

Vol. 19, nr 1

89--101

EN

In this paper, we investigate the existence, uniqueness and Ulam-Hyers stability of solutions for nonlinear implicit fractional differential equations with boundary conditions involving a ψ-Caputo fractional derivative. The obtained results for the proposed problem are proved under a new approach and minimal assumptions on the function ƒ. The analysis is based upon the reduction of the problem considered to the equivalent integral equation, while some fixed point theorems of Banach and Schauder and generalized Gronwall inequality are employed to obtain our results for the problem at hand. Finally, the investigation is illustrated by providing a suitable example.

7

Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral

Bekkouche M. Moumen, Guebbai H.

Journal of Applied Mathematics and Computational Mechanics

|

2020

|

Vol. 19, nr 2

31-42

EN

We study the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo type, which increases the interest of this study. In order to study this problem we have introduced a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo, therefore, the proofs are based upon the reduction of the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and we have built the minimum conditions to obtain the existence and uniqueness of this solution. The analytical study is followed by a complete numerical study.

8

A new modification of the reduced differential transform method for nonlinear fractional partial differential equations

Khalouta Ali, Kadem Abdelouahab

Journal of Applied Mathematics and Computational Mechanics

|

2020

|

Vol. 19, nr 3

45--58

EN

The objective of this study is to present a new modification of the reduced differential transform method (MRDTM) to find an approximate analytical solution of a certain class of nonlinear fractional partial differential equations in particular, nonlinear time-fractional wave-like equations with variable coefficients. This method is a combination of two different methods: the Shehu transform method and the reduced differential transform method. The advantage of the MRDTM is to find the solution without discretization, linearization or restrictive assumptions. Three different examples are presented to demonstrate the applicability and effectiveness of the MRDTM. The numerical results show that the proposed modification is very effective and simple for solving nonlinear fractional partial differential equations.

9

Application of LDG scheme to solve semi-differential equations

Izadi Mohammad

Journal of Applied Mathematics and Computational Mechanics

|

2019

|

Vol. 18, nr 4

27--39

EN

In the current work, we investigate a technique based on discontinuous Galerkin method for the numerical approximation of semi-differential equations with Caputo’s fractional derivative. In this approach, using the natural upwind fluxes enables us to solve the model problem element by element locally in each subintervals and there is no need to solve a full global matrix. Numerical experiments are given to verify the efficiency and accuracy of the proposed method. Numerical solutions are compared with the exact solutions as well as the numerical solutions obtained by other available well-established computational procedures. The results show that the LDG method is more accurate for solving this class of differential equation with relatively low degrees of polynomials and number of elements.

10

Existence and stability of impulsive coupled system of fractional integrodifferential equations

Zada Akbar, Waheed Hira, Alzabut Jehad, Wang Xiaoming

Demonstratio Mathematica

|

2019

|

Vol. 52, nr 1

296--335

EN

In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

11

Nonlinear fractional differential equations with non-instantaneous impulses in Banach spaces

Benchohra M., Slimane M.

Journal of Mathematics and Applications

|

2018

|

Vol. 41

39--51

EN

This paper is devoted to study the existence of solutions for a class of initial value problems for non-instantaneous impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon Monch's fixed point theorem and the technique of measures of noncompactness.

12

Existence and convergence results for Caputo fractional Volterra integro-differential equations

Hamoud A. A., Bani Issa M. Sh., Ghadle K. P., Abdulghani M.

Journal of Mathematics and Applications

|

2018

|

Vol. 41

109--121

EN

In this article, homotopy analysis method is successfully applied to find the approximate solution of Caputo fractional Volterra integro-differential equation. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Also, the behavior of the solution can be formally determined by analytical approximate. Moreover, we proved the existence and convergence of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

13

The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption

Povstenko Y., Klekot J.

Journal of Applied Mathematics and Computational Mechanics

|

2017

|

Vol. 16, nr 2

101--112

EN

The time-fractional heat conduction equation with heat absorption proportional to temperature is considered in the case of central symmetry. The fundamental solutions to the Cauchy problem and to the source problem are obtained using the integral transform technique. The numerical results are presented graphically.

14

Ulam stabilities for the Darboux problem for partial fractional differential inclusions

Abbas S., Benchohra M.

Demonstratio Mathematica

|

2014

|

Vol. 47, nr 4

826--838

EN

In this article, we investigate some Ulam’s type stability concepts for the Darboux problem of partial fractional differential inclusions with a nonconvex valued right hand side. Our results are based upon Covitz-Nadler fixed point theorem and fractional version of Gronwall’s inequality.