Laplace Adomian Decomposition Method Research Articles

Simulation of unsteady MHD flow of incompressible fluid between two parallel plates using Laplace-Adomian decomposition method

The aim of this study is to numerically obtain the solution to an unsteady incompressible fluid confined between two parallel plates with constant velocity. The nonlinear partial differential equation governing the flow is first converted to a nonlinear ordinary differential equation using similarity transformation and solved using the combination of Laplace transform and Adomian decomposition method (LADM). The model parameters affecting the flow geometry is analysed and presented in tables and graphs. The method produces an accurate, elegant, computable, and approximately convergent solution which gives great insight but with less mathematical sophistication.

Simulation of unsteady MHD flow of incompressible fluid between two parallel plates using Laplace-Adomian decomposition method

Simulation of unsteady MHD flow of incompressible fluid between two parallel plates using Laplace-Adomian decomposition method

The Integral Iterative Method for Approximate Solution of Newell-Whitehead-Segel Equation

In this paper, the Newell-Whitehead-Segel (NWS) equation is solved using the integral iterative method (IIM) to determine the accuracy and effectiveness of the method. Comparison of results obtained by IIM with the exact solution and other existing results obtained by other methods such as new iterative method (NIM), Adomian decomposition method (ADM) and Laplace Adomian decomposition method (LADM) revealed the accuracy and effectiveness of the method. The approximation results obtained by IIM is comparable with the others. IIM is reliable and easier in solving the nonlinear problems since this method is simple, straightforward and does not require calculating multiple integral and demand less computational work.

Mathematical analysis of Hepatitis C Virus infection model in the framework of non-local and non-singular kernel fractional derivative

In this paper, we study a mathematical model of Hepatitis C Virus (HCV) infection. We present a compartmental mathematical model involving healthy hepatocytes, infected hepatocytes, non-activated dendritic cells, activated dendritic cells and cytotoxic T lymphocytes. The derivative used is of non-local fractional order and with non-singular kernel. The existence and uniqueness of the system is proven and its stability is analyzed. Then, by applying the Laplace Adomian decomposition method for the fractional derivative, we present the semi-analytical solution of the model. Finally, some numerical simulations are performed for concrete values of the parameters and several graphs are plotted to reveal the qualitative properties of the solutions.

The generalized time-fractional Fornberg–Whitham equation: An analytic approach

This work implements a novel analytical approach known as the q-Homotopy Analysis Shehu Transform Method to the time-fractional Fornberg–Whitham equation in Caputo’s sense. The proposed method combines the idea of the q-Homotopy analysis method with the Shehu transform to enhance the efficacy of the proposed method. To demonstrate the correctness of the proposed method, the convergence analysis of the method has been obtained along with its term approximations. Finally, the obtained numerical solution is compared with the available Laplace Adomian decomposition method (LADM) solution and with the exact answer to test the efficiency of the q-HAShTM through the control parameter ħ and n.

Conceptual Investigation of the Disease Transmission Coefficient in Seir Epidemic Model Using Laplace Adomian Decomposition Method (LADM)

In this paper, the impact of disease transmission coefficient on a SEIR epidemic model using Laplace Adomian Decomposition Method (LADM) towards disease eradication is presented. Numerical Simulations which show the effect of transmission coefficient are shown with the use of Maple 18 and the results discussed extensively. The simulation results show that disease transmission coefficient plays vital role in disease eradication.

Free vibration analysis of non-uniform Bernoulli beam by using Laplace Adomian decomposition method

This paper discusses an investigation into the influence of the cone ratio and axial force on the vibration problem in a non-uniform cantilever Euler–Bernoulli beam. In the analysis, the governing equation for the non-uniform cantilever Euler–Bernoulli beam is solved using the Laplace Adomian decomposition method (LADM). The LADM is used to convert the governing equation into a characteristic equation of a non-uniform Euler–Bernoulli beam, and a simple calculation is performed to obtain the natural frequencies and corresponding modals. The obtained numerical results are verified using a comparison with the results reported in previous studies. The present study speeds up the convergent rate and the accuracy of calculation by comparing the results using the modified Adomian decomposition method (MADM) and differential transformation method (DTM). The main power and advantage of the LADM are providing an analytical approximation to a nonlinear differential equation without linearization, perturbation, approximation, and discretization, all of which lead to huge numerical computation. The numerical methods demonstrated that the natural frequency increases with increasing the rotating spring modulus and moving spring modulus, and the moving spring modulus has a greater influence on the natural frequency. The effects of the cone ratio and axial force are presented for non-uniform Euler–Bernoulli beams. The numerical results show that the LADM is a suitable technique for analyzing the behavioral characteristics of beams.

A fractional order model for Dual Variants of COVID-19 and HIV co-infection via Atangana-Baleanu derivative

In this paper, a new mathematical model for dual variants of COVID-19 and HIV co-infection is presented and analyzed. The existence and uniqueness of the solution of the proposed model have been established using the well known Banach fixed point theorem. The model is solved semi-analytically using the Laplace Adomian decomposition Method. The impact of the Atangana-Baleanu fractional derivative on the dynamics of the proposed model is studied. The work also highlights the impact of COVID-19 vaccination on the dynamics of the co-infection of both diseases. The model is fitted to real COVID-19 data from Botswana. The impact of COVID-19 variants on HIV prevalence using simulations is also assessed. Simulation for the class of individuals co-infected with HIV and the wild or Delta COVID-19 variant reveals a significant decrease, as vaccination rate is increased. The impact of fractional order on different epidemiological classes is also studied. Drawing the plot of total infected population with the wild and Delta COVID-19 variants, at different vaccination rates, it is concluded that, as vaccination rate is increased, there is a significant reduction in population infected with the wild and delta COVID-19 variants. The plot of class of individuals co-infected with HIV and the wild or Delta COVID-19 variant is more interesting; as vaccination rate is increased, the co-infected populations experience a significant decrease. Thus, stepping up vaccination against the different variants of COVID-19 could reduce co-infection cases largely, among people already infected with HIV.

Laplace decomposition method for series solutions of systems of linear partial differential models

Most of the real-life problems experienced in the industry these days cannot be expressed as a single differential equation but as a system of differential equations. Such structures of linear partial differential models are considered in this work with the aid of the Laplace Adomian decomposition method (LADM) in terms of solutions. Various examples are taken into consideration. The results are easily obtained, are in good agreement when compared to their exact forms. In addition, the proposed method seems effective and efficient; the solutions are graphically presented.

Study of the Fractional-Order HIV-1 Infection Model with Uncertainty in Initial Data

Uncertainty always lives with us. We cannot take exact measurement of initial conditions or parameters values in a mathematical model. As humans, we are remaining alive in an environment where the uncertainties lie in the modelling of physical phenomena. There might be some incomplete information or estimation of the parameter or initial values. To handle uncertainty, we use fuzzy operators rather than classical operators. In this paper, we study a model of HIV-1 infection by taking uncertainty in the initial data under Caputo fractional operator. We explore the existence and uniqueness of the results through fixed-point theory. We study the Ulam–Hyres stability of the considered model. By using the fuzzy Laplace Adomian decomposition method, numerical results are obtained for specific fuzzy initial conditions. To better understand the behaviour of the fuzzy solution, we present the obtained numerical results graphically for various fractional orders where the uncertainty lies in [0, 1].

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 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>

Nonlinear analysis of a nonlinear modified KdV equation under Atangana Baleanu Caputo derivative

<abstract><p>The focus of the current manuscript is to provide a theoretical and computational analysis of the new nonlinear time-fractional (2+1)-dimensional modified KdV equation involving the Atangana-Baleanu Caputo ($ \mathcal{ABC} $) derivative. A systematic and convergent technique known as the Laplace Adomian decomposition method (LADM) is applied to extract a semi-analytical solution for the considered equation. The notion of fixed point theory is used for the derivation of the results related to the existence of at least one and unique solution of the mKdV equation involving under $ \mathcal{ABC} $-derivative. The theorems of fixed point theory are also used to derive results regarding to the convergence and Picard's X-stability of the proposed computational method. A proper investigation is conducted through graphical representation of the achieved solution to determine that the $ \mathcal{ABC} $ operator produces better dynamics of the obtained analytic soliton solution. Finally, 2D and 3D graphs are used to compare the exact solution and approximate solution. Also, a comparison between the exact solution, solution under Caputo-Fabrizio, and solution under the $ \mathcal{ABC} $ operator of the proposed equation is provided through graphs, which reflect that $ \mathcal{ABC} $-operator produces better dynamics of the proposed equation than the Caputo-Fabrizio one.</p></abstract>

Galerkin time discretization scheme for the transmission dynamics of HIV infection with non-linear supply rate

<abstract><p>The present work implements the continuous Galerkin-Petrov method (cGP(2)-method) to compute an approximate solution of the model for HIV infection of $ \text{CD4}^{+} $ T-cells. We discuss and analyse the influence of different clinical parameters on the model. The work also depicts graphically that how the level of $ \text{CD4}^{+} $ T-cells varies with respect to the emerging parameters in the model. Simultaneously, the model is solved using the fourth-order Runge Kutta (RK4) method. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with those obtained through the RK4 method. A numerical comparison between the results of the cGP (2) method and the RK4 method reveals that the proposed technique is a promising tool for the approximate solution of non-linear systems of differential equations. The present study highlights the accuracy and efficiency of the proposed schemes as in comparison to the other traditional schemes, for example, the Laplace adomian decomposition method (LADM), variational iteration method (VIM), homotopy analysis method (HAM), homotopy perturbation method (HAPM), etc. In this study, two different versions of the HIV model are considered. In the first one, the supply of new $ \text{CD4}^{+} $ T-cells from the thymus is constant, while in the second, we consider the production of these cells as a monotonically decreasing function of viral load. The experiments show that the lateral model provides more reasonable predictions than the former model.</p></abstract>

Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data.

In this paper, we construct the SV1V2EIR model to reveal the impact of two-dose vaccination on COVID-19 by using Caputo fractional derivative. The feasibility region of the proposed model and equilibrium points is derived. The basic reproduction number of the model is derived by using the next-generation matrix method. The local and global stability analysis is performed for both the disease-free and endemic equilibrium states. The present model is validated using real data reported for COVID-19 cumulative cases for the Republic of India from 1 January 2022 to 30 April 2022. Next, we conduct the sensitivity analysis to examine the effects of model parameters that affect the basic reproduction number. The Laplace Adomian decomposition method (LADM) is implemented to obtain an approximate solution. Finally, the graphical results are presented to examine the impact of the first dose of vaccine, the second dose of vaccine, disease transmission rate, and Caputo fractional derivatives to support our theoretical results.

On the fractional-order mathematical model of COVID-19 with the effects of multiple non-pharmaceutical interventions

<abstract><p>In this article, the Caputo fractional derivative operator of different orders $ 0 < \alpha\leq1 $ is applied to formulate the fractional-order model of the COVID-19 pandemic. The existence and boundedness of the solutions of the model are investigated by using the Gronwall-Bellman inequality. Further, the uniqueness of the model solutions is established by using the fixed-point theory. The Laplace Adomian decomposition method is used to obtain an approximate solution of the nonlinear system of fractional-order differential equations of the model with a different fractional-order $ \alpha $ for every compartment in the model. Finally, graphical presentations are presented to show the effects of other fractional parameters $ \alpha $ on the obtained approximate solutions.</p></abstract>

Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels

<abstract><p>In this article, we study the nonlinear sine-Gordon equation (sGE) under Mittag-Leffler and exponential decay type kernels in a fractal fractional sense. The Laplace Adomian decomposition method (LADM) is applied to investigate the sGE under the above-mentioned operators. The convergence analysis is provided for the proposed method. The results are validated by considering numerical examples with different initial conditions for both kernels and confirm the competence of the proposed technique. It is revealed that the obtained series solutions of the model with fractal fractional operators converge to the exact solutions. The numerical results converge to the particular exact solutions, proving the significance of LADM for nonlinear systems under fractal fractional derivatives. The absolute error analysis between the exact and obtained series solutions with both operators is shown in the tabulated form. The physical interpretations of the attained results with different fractal and fractional parameters are discussed in detail.</p></abstract>

Analytical Approach to Fractional Fisher Equations by Laplace-Adomian Decomposition Method

This article implements the Laplace-Adomian decomposition method to obtain approximate analytical solutions in series form for non-linear time-fractional Fisher's equations with initial conditions. The fractional derivatives are given in the sense of Caputo. In addition, the results of this investigation are represented graphically, and they are simple yet highly accurate and compare favourably with the solutions reported in the earliest literature.

Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels

<abstract><p>This article presents the Laplace-Adomian decomposition method (LADM), which produces a fast convergence series solution, for two types of nonlinear fractional Sturm-Liouville (SL) problems. The fractional derivatives are defined in the Caputo, conformable, Caputo-Fabrizio in the sense of Caputo (CFC), Caputo type Atangana-Baleanu (ABC) senses. With the help of this method, approximate solutions of the investigated problems were obtained. The solutions generated from the Caputo and ABC derivatives are represented by the Mittag-Leffler function, which is intrinsic to fractional derivatives, and the solution obtained using the conformable and CFC derivatives generate the hyperbolic sine and cosine functions. Thus, we derive some novel solutions for fractional-order versions of nonlinear SL equations. The fractional calculus provides more data than classical calculus and has been widely used in mathematical modeling with memory effect. Finally, we analyzed and compared these novel solutions of the considered problems by graphs under different values of $ p $, $ \lambda $ and different orders of $ \alpha $.</p></abstract>

On the approximate numerical solutions of fractional heat equation with heat source and heat loss

In this paper, we are interested in obtaining an approximate numerical solution of the fractional heat equation where the fractional derivative is in Caputo sense. We also consider the heat equation with a heat source and heat loss. The fractional Laplace-Adomian decomposition method is applied to gain the approximate numerical solutions of these equations. We give the graphical representations of the solutions depending on the order of fractional derivatives. Maximum absolute error between the exact solutions and approximate solutions depending on the fractional-order are given. For the last thing, we draw a comparison between our results and found ones in the literature.