Fractional Differential Equations

A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods

In this endeavour, Bernstein wavelet and Euler methods are used to solve a nonlinear fractional predator-prey biological model of two species. The theoretical results with their corresponding biological consequence due to Bernstein wavelet are considered and discussed. A test problem of predator-prey model with two different cases are examined to determined the capability of our proposed methods. We showed that the obtained solutions are the most powerful and, wherever it is possible the comparison, in a very good coincidence with the other numerical solution. Few numerical simulations are finding for predator and prey populations and new chaotic behaviours of predator-prey population model are also obtained by using the Euler method. Moreover, a comparison have been done between the capability of the Bernstein wavelet and the Euler approach. The numerical simulations and behaviours of Rabies model are depicted through graphically which is a special case of predator-prey model.

Chaotic behaviour of fractional predator-prey dynamical system

By using the fractional Caputo–Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the equation by a combination of the Laplace transform and homotopy analysis method. Finally, we provide some numerical analytics and comparisons of the results.

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Analysis of the model of HIV-1 infection of CD4+$CD4^{+}$ T-cell with a new approach of fractional derivative

An efficient analytical technique for fractional model of vibration equation

In this paper, we constitute a homotopy algorithm basically extension of homotopy analysis method with Laplace transform, namely q-homotopy analysis transform method to solve time- and space-fractional coupled Burgers’ equations. The suggested technique produces many more opportunities by appropriate selection of auxiliary parameters ℏ and n ( n ⩾ 1 ) to solve strongly nonlinear differential equations. The proposed technique provides ℏ and n -curves, which describe that the convergence range is not a local point effects and finds elucidated series solution that makes it superior than HAM and other analytical techniques.

Numerical solution of time- and space-fractional coupled Burgers’ equations via homotopy algorithm

In this work, we aim to present a hybrid numerical scheme based on the homotopy analysis transform method (HATM) to examine the fractional model of nonlinear wave-like equations having variable coefficients, which narrate the evolution of stochastic systems. The wave-like equation models the erratic motions of small particles that are dipped in fluids and fluctuations of the stochastic behavior of exchange rates. The uniqueness and existence of HATM solution have also been discussed. Some numerical examples are given to establish the accurateness and effectiveness of the suggested scheme. Furthermore, we show that the proposed computational approach can give much better approximation than perturbation and Adomain decomposition method, which are the special cases of HATM. The result exhibits that the HATM is very productive, straight out and computationally very attractive.

A hybrid analytical algorithm for nonlinear fractional wave-like equations

In this work, we concentrate on the analysis of the time-fractional Rosenau–Hyman equation occurring in the formation of patterns in liquid drops via q-homotopy analysis transform technique and reduced differential transform approach. The q-homotopy analysis transform algorithm can provide rapid convergent series by choosing the appropriate values of auxiliary parameters ħ and n at large domain. The reduced differential transform technique gives wider applicability due to reduction in computations and makes the calculation much simpler and easier. The proposed techniques are realistic and free from any assumption and perturbation for solving the time-fractional Rosenau–Hyman equation.

An efficient computational approach for time-fractional Rosenau–Hyman equation

In this article, we implemented homotopy transform methods, namely, homotopy analysis transform method and homotopy perturbation Sumudu transform method to examine the fractional model for HIV infection of CD4+T lymphocyte cells. Proliferation of CD4+T lymphocyte cells is driven by a combination of the homeostatic response to cells depletion (CD4+T cells counts) and viral load (HIV levels). The attraction of both the methods is that an approach of HPSTM is used and on other hand by HATM a large admissible convergence range of series solution is described for standard as well as fractional order nonlinear problems.

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Analytic study for a fractional model of HIV infection of C D 4 + T lymphocyte cells

Abstract Empirical investigations of solute fate and carrying in streams and rivers often contain inventive liberate of solutes at an upstream perimeter for a finite interval of time. An analysis of various worth references on surface-water-grade mathematical formulation reveals that the logical solution to the continual-parameter advection- dispersion problem for this type of boundary state has been generally missed. In this work, we study the q-fractional homotopy analysis transform method (q-FHATM) to find the analytical and approximate solutions of space-time arbitrary order advection-dispersion equations with nonlocal effects. The diagrammatical representation is done by using Maple package, which enhance the discretion and stability of family of q-FHATM series solutions of fractional advection-dispersion equations. The efficiency of the applied technique is demonstrated by using three numerical examples of space- and time-fractional advection-dispersion equations.

Ram Swroop

Papers

Numerical solution of time- and space-fractional coupled Burgers’ equations via homotopy algorithm

A hybrid analytical algorithm for nonlinear fractional wave-like equations

An efficient computational approach for time-fractional Rosenau–Hyman equation

Analytic study for a fractional model of HIV infection of C D 4 + T lymphocyte cells

A reliable analytical approach for a fractional model of advection-dispersion equation