Homotopy Decomposition Method Research Articles

Homotopy decomposition method to analysis fractional hepatitis B virus infection model

Mathematical models have been employed to investigate the factors that govern the progression of infectious diseases in viral infections. This article investigates a fractionalised model for hepatitis B virus infection, curing infected cells. We analysed the fractional hepatitis B virus infection model using the homotopy decomposition method (HDM). Additionally, the existence and uniqueness of the fractional model are considered. The graphical solutions depicted that when introducing a fractional-order derivative to the model, the peak of viral propagation is reduced, and cell damage is minimised by initiating treatment at an early stage; however, the disease takes a longer time to be eradicated. The outcomes may also be valuable in the fields of clinical science and medicine. This model provided insights into the natural history of HBV infection, including the different stages of infection (acute, chronic, and resolved) and the likelihood of progression from one stage to another. This information is essential for predicting disease outcomes and planning appropriate clinical management for patients.

A new analytical approximate solution of fractional coupled Korteweg-de Vries system

The main objective of this work is to present a modification of the Mittag- Leffler function to deduce a relatively new analytical approximate method (for short MMLFM) able to solve time-fractional nonlinear partial differential equations (PDEs). Moreover, we employ the MMLFM to solve the time-fractional coupled Korteweg?de Vries (KdV) model described by two nonlinear fractional partial differential equations (FPDEs) based upon Caputo fractional derivative (CFD). The simulation of projected results is presented in some figures and tables. Furthermore, we compare our solutions when ? = 1 with known exact solutions which indicate a good agreement, in addition, we compare our outcomes with the results obtained by other methods in the literature such as the Natural decomposing method (NDM) and homotopy decomposition method (HDM) in order to prove the reliability and efficiency of our used method. Also, we display solutions with different values of ? to present the effect of the fractional order on the proposed problem. The results of this article reveal the advantages of the MMLFM, which is simple, reliable, accurate, needs simple mathematical computations, is rapidly convergent to the exact solution, have a straightforward and easy algorithm compared to other analytical methods to study linear and nonlinear FPDEs, which makes this technique suited for real industrial or medical applications.

On the existence and uniqueness analysis of fractional blood glucose-insulin minimal model

In our research work, we suggest the modified minimal model of fractional order and analyze it using the homotopy decomposition method (HDM). The minimal model is quite a useful mathematical model which describes the behavior of glucose-insulin metabolism. The original model was given in the 80s and has been updated over a different period. In this modified model, we add a one-factor diet which plays an important role in the blood-glucose analysis. We obtained the numerical results by using the homotopy decomposition method. HDM is extremely useful, significant, and very simple. We also discuss the existence and uniqueness of the fractional model.

Application of a hybrid method for systems of fractional order partial differential equations arising in the model of the one-dimensional Keller-Segel equation

In this paper, we apply a hybrid method due to coupling the Laplace transform with the Adomian decomposition method (LADM) for solving nonlinear fractional differential equations that appear in the model of Keller-Segel equations with one dimension. We explain the adopted method is with several examples. It turns out that the reliability of LADM and the reductions in computations show that LADM is widely applicable. We also compare our results with the results of homotopy decomposition method (HDM).

A study of the fractional‐order mathematical model of diabetes and its resulting complications

Diabetes is a worldwide problem that affects one of every 11 persons nowadays. The IDF Diabetes Atlas (Eighth edition, 2017) states that approximately 415 million people in the world are living with the disease and that this number will rise to 629 million by the year 2045. It is a very serious problem of the world. A major part of the world population is affected by this disease and its resulting complications. In this paper, we propose to investigate a fractional‐order model of diabetes and its resulting complications. The mathematical model's parameters define the population of diabetic patients and those who are diabetic with complications at a given time t. We have also discussed the existence, uniqueness, and stability of the fractional‐order model, which we consider here. We make use of the homotopy decomposition method (HDM) in order to solve the problem.

Homotopy Decomposition Method for Solving Higher-Order Time- Fractional Diffusion Equation via Modified Beta Derivative

In this paper, the homotopy decomposition method with a modified definition of beta fractional derivative is adopted to find approximate solutions of higher-dimensional time-fractional diffusion equations. To apply this method, we find the modified beta integral for both sides of a fractional differential equation first, then using homotopy decomposition method we can obtain the solution of the integral equation in a series form. We compare the solutions obtained by the proposed method with the exact solutions obtained using fractional variational homotopy perturbation iteration method via modified Riemann-Liouville derivative. The comparison shows that the results are in a good agreement.

Metric-like spaces to prove existence of solution for nonlinear quadratic integral equation and numerical method to solve it

The purpose of this paper is to obtain some common fixed point results for two mappings satisfying various contractive conditions in metric-like spaces. These results extend some previous results in the literature, since the condition under which the operator admits common fixed points is more general than the others in literature. Therefore, several well known results are generalized. As an application we use these results to existence of solution for nonlinear quadratic integral equation. To credibility, we apply modified homotopy and Adomian decomposition method to find solution of the above problem with high accuracy.

Approximate solution of cryptosporidiosis model

Cryptosporidium is associated with waterborne transmission mechanism through the faecal–oral path in many recreational water facilities. We investigate the probable approximate solution of integer and noninteger systems of nonlinear ordinary differential equations representing cryptosporidiosis dynamics. The approximate or estimate solution is derived through recent developed analytic method, the homotopy decomposition method (HDM). The algorithm is systemically explained and demonstrated with some numerical examples. The numerical results indicate that the approximate solution is of continuous function form in the light of noninteger-order derivative. The integer-order numerical solution of parameters values varied and investigated which show similar solution in each case. The method employed to obtain the solution to this problem is robust, easy, reliable and quick in terms of time.

Performance Characterization of Quantum Dot Solar Cell using Homotopy Decomposition Method

The 3D numerical model of InGaAs quantum dot solar cell is developed and presented in this paper. The device characteristics are examined based on the exact potential and energy profile of the quantum dot obtained from the solution of 3D Poisson and Schrodinger equations using Homotopy decomposition method. The QD current is estimated by considering the QD parameters and the results obtained show that the QD current is strongly influenced by quantum dot density and applied voltage. The characteristics of the quantum dot solar cell such as photocurrent, spectral response, recombination rate for the quantum wavelength and QD layers are obtained for various device parameters. The numerical simulation results exhibit the strength of the proposed model.

Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal

Abstract In this work, a theoretical study of diffusion of neumatic liquid crystals was done using the concept of fractional order derivative. This version of fractional derivative is very easy to handle and obey to almost all the properties satisfied by the conventional Newtonian concept of derivative. The mathematical equation underpinning this physical phenomenon was solved analytically via the so-called homotopy decomposition method. In order to show the accuracy of this iteration method, we constructed a Hilbert space in which we proved its stability for the time-fractional Hunder-Saxton equation.

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

On the stability of iteration methods for special solution of time-fractional generalized nonlinear ZK-BBM equation

An investigation of a possible description of non-linear wave was performed inside the scope of fractional order derivative. A clear motivation underpinning the extension of this study within the concept of fractional calculus was presented. The special solution of the extended equation was derived using an efficient iteration method called homotopy decomposition method. A theorem underpinning the uniqueness of the special solution was proposed and proved in detail. An investigation for stability of the iteration method for solving the extended equation was presented in detail. Numerical simulations were carried out using the proposed algorithm for different values of alpha and n.

Computational analysis of the model describing HIV infection of CD4+T Cells.

An analysis of the model underpinning the description of the spread of HIV infection of CD4+T cells is examined in detail in this work. Investigations of the disease free and endemic equilibrium are done using the method of Jacobian matrix. An iteration technique, namely, the homotopy decomposition method (HDM), is implemented to give an approximate solution of nonlinear ordinary differential equation systems. The technique is described and illustrated with numerical examples. The approximated solution obtained via HDM is compared with those obtained via other methods to prove the trustworthiness of HDM. Moreover, the lessening and simplicity in calculations furnish HDM with a broader applicability.

Modelling the aggregation process of cellular slime mold by the chemical attraction.

We put into exercise a comparatively innovative analytical modus operandi, the homotopy decomposition method (HDM), for solving a system of nonlinear partial differential equations arising in an attractor one-dimensional Keller-Segel dynamics system. Numerical solutions are given and some properties show evidence of biologically practical reliance on the parameter values. The reliability of HDM and the reduction in computations give HDM a wider applicability.

On the Singular Perturbations for Fractional Differential Equation

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.

On the Hamilton-Jacobi-Bellman Equation by the Homotopy Perturbation Method

Our concern in this paper is to use the homotopy decomposition method to solve the Hamilton-Jacobi-Bellman equation (HJB). The approach is obviously extremely well organized and is an influential procedure in obtaining the solutions of the equations. We portrayed particular compensations that this technique has over the prevailing approaches. We presented that the complexity of the homotopy decomposition method is of orderO(n). Furthermore, three explanatory examples established good outcomes and comparisons with exact solution.

On the Fractional Nagumo Equation with Nonlinear Diffusion and Convection

We presented the Nagumo equation using the concept of fractional calculus. With the help of two analytical techniques including the homotopy decomposition method (HDM) and the new development of variational iteration method (NDVIM), we derived an approximate solution. Both methods use a basic idea of integral transform and are very simple to be used.

Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations

In this paper, we make use of the relatively new analytical technique, the homotopy decomposition method (HDM), to solve a system of fractional nonlinear differential equations that arise in the model for HIV infection of CD4+ T cells and attractor one-dimensional Keller-Segel equations. The technique is described and illustrated with a numerical example. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward.

Solving Partial Differential Equation with Space- and Time-Fractional Derivatives via Homotopy Decomposition Method

The analytical solution of the partial differential equation with time- and space-fractional derivatives was derived by means of the homotopy decomposition method (HDM). Some examples are given and comparisons are made. The evaluations show that the homotopy decomposition method is extremely successful and suitable. The achieved results make the steadfastness of the HDM and its wider applicability to fractional differential equation obvious. Additionally, the adding up implicated in HDM is exceptionally undemanding and uncomplicated. It is confirmed that HDM is an influential and professional apparatus for FPDEs. It was also established that HDM is supplementary well organized than the ADM, VIM, HAM, and HPM.

Analytical Solutions of Boundary Values Problem of 2D and 3D Poisson and Biharmonic Equations by Homotopy Decomposition Method

The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. The method does not require any corrected function or any Lagrange multiplier and it avoids repeated terms in the series solutions compared to the existing decomposition method including the variational iteration method, the Adomian decomposition method, and Homotopy perturbation method. The approximated solutions obtained converge to the exact solution as tends to infinity.