Efficient Approaches for Solving Systems of Nonlinear Time-Fractional Partial Differential Equations

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions.

Numerical Simulation of Cubic-Quartic Optical Solitons with Perturbed Fokas–Lenells Equation Using Improved Adomian Decomposition Algorithm

The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known results in the literature. As proven, the proposed scheme is efficient and possess a high level of accuracy.

Numerical simulation of time partial fractional diffusion model by Laplace transform

<abstract><p>In the present work, the authors developed the scheme for time Fractional Partial Diffusion Differential Equation (FPDDE). The considered class of FPDDE describes the flow of fluid from the higher density region to the region of lower density, macroscopically it is associated with the gradient of concentration. FPDDE is used in different branches of science for the modeling and better description of those processes that involve flow of substances. The authors introduced the novel concept of fractional derivatives in term of both time and space independent variables in the proposed FPDDE. We provided the approximate solution for the underlying generalized non-linear time PFDDE in the sense of Caputo differential operator via Laplace transform combined with Adomian decomposition method known as Laplace Adomian Decomposition Method (LADM). Furthermore, we established the general scheme for the considered model in the form of infinite series by aforementioned techniques. The consequent results obtained by the proposed technique ensure that LADM is an effective and accurate technique to handle nonlinear partial differential equations as compared to the other available numerical techniques. At the end of this paper, the obtained numerical solution is visualized graphically by Matlab to describe the dynamics of desired solution.</p></abstract>

Modelling of natural growth with memory effect in economics: An application of adomian decomposition and variational iteration methods

The power-law memory effect is taken into consideration in a generalisation of the economic model of natural growth. The memory effect refers to a process's reliance on its current state and its history of previous changes. However, the study that focuses on natural growth in economics considering the memory effect with fractional order-linear differential equation model is still limited. The current investigation seeks to solve the natural growth with memory effect in the economics model and decide the best model using fractional differential equation (FDE), namely Adomian Decomposition and Variational Iteration Methods. Also, this study assumes the level of consumer loss memory during a certain time interval denoted by a parameter (α). This study showed the model of loss memory effect with 0 < α ≤ 1 given a slowdown in output growth compared to a model without memory effect. Besides that, this study also found that output Y(t) is growing faster with the Variational Iteration method compared to the Adomian decomposition method. Also, using graphical simulation, this study found the output Y(t) is closer to the exact solution with α=0.4 and α=0.9. In conclusion, this study successfully solved natural growth with memory effect in economics and decided the best model between FDE, namely Adomian decomposition and Variational iterative methods using numerical analysis.

Solution Non Linear Partial Differential Equations By ZMA Decomposition Method

In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

Analytical Analysis of Fractional-Order Physical Models via a Caputo-Fabrizio Operator

This paper investigates a modified analytical method called the Adomian decomposition transform method for solving fractional-order heat equations with the help of the Caputo-Fabrizio operator. The Laplace transform and the Adomian decomposition method are implemented to obtain the result of the given models. The validity of the proposed method is verified by considering some numerical problems. The solution achieved has shown that the better accuracy of the suggested method. Furthermore, due to the straightforward implementation, the proposed method can solve other nonlinear fractional-order problems.

Numerical, Approximate Solutions, and Optimal Control on the Deathly Lassa Hemorrhagic Fever Disease in Pregnant Women

This paper is devoted to the model of Lassa hemorrhagic fever (LHF) disease in pregnant women. This disease is a biocidal fever and epidemic. LHF disease in pregnant women has negative impacts that were initially appeared in Africa. In the present study, we find an approximate solution to the fractional-order model that describes the fatal LHF disease. Laplace transforms coupled with the Adomian decomposition method (ADM) are applied. In addition, the fractional-order LHF model is numerically simulated in terms of a varied fractional order. Furthermore, a fractional order optimal control for the LHF model is studied.

Solving Generalized Nonlinear Schrödinger Equation by Adomian Decomposition Technique

In this paper, using a suitable change of variable and applying the Adomian decomposition method to the generalized nonlinear Schr¨odinger equation, we obtain the analytical solution, taking into account the parameters such as the self-steepening factor, the second-order dispersive parameter, the third-order dispersive parameter and the nonlinear Kerr effect coefficient, for pulses that contain just a few optical cycle. The analytical solutions are plotted. Under influence of these effects, pulse did not maintain its initial shape.

On Comparative Analysis for the Black-Scholes Model in the Generalized Fractional Derivatives Sense via Jafari Transform

The Black-Scholes model is well known for determining the behavior of capital asset pricing models in the finance sector. The present article deals with the Black-Scholes model via the Caputo fractional derivative and Atangana-Baleanu fractional derivative operator in the Caputo sense, respectively. The Jafari transform is merged with the Adomian decomposition method and new iterative transform method. It is worth mentioning that the Jafari transform is the unification of several existing transforms. Besides that, the convergence and uniqueness results are carried out for the aforesaid model. In mathematical terms, the variety of equations and their solutions have been discovered and identified with various novel features of the projected model. To provide additional context for these ideas, numerous sorts of illustrations and tabulations are presented. The precision and efficacy of the proposed technique suggest its applicability for a variety of nonlinear evolutionary problems.

Analytical Investigation of Noyes–Field Model for Time-Fractional Belousov–Zhabotinsky Reaction

In this article, we find the solution of time-fractional Belousov–Zhabotinskii reaction by implementing two well-known analytical techniques. The proposed methods are the modified form of the Adomian decomposition method and homotopy perturbation method with Yang transform. In Caputo manner, the fractional derivative is used. The solution we obtained is in the form of series which helps in investigating the analytical solution of the time-fractional Belousov–Zhabotinskii (B-Z) system. To verify the accuracy of the proposed methods, an illustrative example is taken, and through graphs, the solution is shown. Also, the fractional-order and integer-order solutions are compared with the help of graphs which are easy to understand. It has been verified that the solution obtained by using the given approaches has the desired rate of convergence to the exact solution. The proposed technique’s principal benefit is the low amount of calculations required. It can also be used to solve fractional-order physical problems in a variety of domains.