Numerical Solution of Fractional-Order HIV Model Using Homotopy Method

In this study, we construct a convergent algorithm for generating an approximate analytic solution for the fractional HIV infection of CD4+ T cells with Atangana–Baleanu fractional derivatives in the Caputo sense. We compute the solution by utilizing the fractional homotopy analysis transform method (FHATM) and achieved a convergence region of the solution by employing an auxiliary parameter. Moreover, we apply a numerical scheme proposed by Toufik and Atangana for solving this kind of problem and compared with our results. A good agreement between the new algorithm and the numerical scheme is remarkable. The solution via the present algorithm can be obtained without any linearization or discretization which makes it reliable and easy to apply.

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On a new modified fractional analysis of Nagumo equation

International Journal of Biomathematics ◽

10.1142/s1793524519500347 ◽

2019 ◽

Vol 12 (03) ◽

pp. 1950034 ◽

Cited By ~ 15

Author(s):

Khaled M. Saad ◽

Si̇nan Deni̇z ◽

Dumi̇tru Baleanu

Keyword(s):

Exact Solution ◽

Approximate Solution ◽

Convergence Analysis ◽

Fractional Derivatives ◽

Transform Method ◽

Average Residual ◽

Homotopy Analysis ◽

Fractional Analysis ◽

Nagumo Equation ◽

Modified Equation

In this work, a new modified fractional form of the Nagumo equation has been presented and deeply analyzed. Using the Caputo–Fabrizio and Atangana–Baleanu time-fractional derivatives, classical Nagumo model is transformed to a new fractional version. The modified equation has been solved by using the homotopy analysis transform method. The convergence analysis has been also examined with the help of the so-called [Formula: see text]-curves and average residual error. Comparing the obtained approximate solution with the exact solution leaves no doubt believing that the proposed technique is very efficient and converges toward the exact solution very rapidly.

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Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method

ISRN Mathematical Physics ◽

10.5402/2012/197068 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11 ◽

Cited By ~ 30

Author(s):

Najeeb Alam Khan ◽

Muhammad Jamil ◽

Asmat Ara

Keyword(s):

Homotopy Analysis Method ◽

Fractional Derivatives ◽

Approximate Solutions ◽

Higher Dimensions ◽

Analysis Method ◽

Transform Method ◽

Coupled Equations ◽

Fractional Schrödinger Equations ◽

Differential Transform ◽

Homotopy Analysis

We construct the approximate solutions of the time-fractional Schrödinger equations, with zero and nonzero trapping potential, by homotopy analysis method (HAM). The fractional derivatives, in the Caputo sense, are used. The method is capable of reducing the size of calculations and handles nonlinear-coupled equations in a direct manner. The results show that HAM is more promising, convenient, efficient and less computational than differential transform method (DTM), and easy to apply in spaces of higher dimensions as well.

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Approximate Analytical Solution for the Forced Korteweg-de Vries Equation

Journal of Applied Mathematics ◽

10.1155/2013/795818 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Vincent Daniel David ◽

Mojtaba Nazari ◽

Vahid Barati ◽

Faisal Salah ◽

Zainal Abdul Aziz

Keyword(s):

Vries Equation ◽

Nonlinear Problems ◽

Analysis Method ◽

Convergence Region ◽

Analytical Technique ◽

De Vries ◽

Homotopy Analysis ◽

And Control ◽

Good Agreement ◽

Fkdv Equation

The forced Korteweg-de Vries (fKdV) equations are solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique which provides a novel way to obtain series solutions of such nonlinear problems. It has the auxiliary parameterℏ, where it is easy to adjust and control the convergence region of the series solution. Some examples of forcing terms are employed to analyse the behaviours of the HAM solutions for the different fKdV equations. Finally, this form of HAM solution is compared with the analytical soliton-type solution of fKdV equation as derived by Zhao and Guo. The results is found to be in good agreement with Zhao and Guo.

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Davey-Stewartson Equation with Fractional Coordinate Derivatives

The Scientific World JOURNAL ◽

10.1155/2013/941645 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

H. Jafari ◽

K. Sayevand ◽

Yasir Khan ◽

M. Nazari

Keyword(s):

Fractional Derivative ◽

Exact Solutions ◽

Fractional Order ◽

Homotopy Analysis Method ◽

Fractional Derivatives ◽

Stability And Convergence ◽

Analysis Method ◽

Homotopy Analysis ◽

Good Agreement

We have used the homotopy analysis method (HAM) to obtain solution of Davey-Stewartson equations of fractional order. The fractional derivative is described in the Caputo sense. The results obtained by this method have been compared with the exact solutions. Stability and convergence of the proposed approach is investigated. The effects of fractional derivatives for the systems under consideration are discussed. Furthermore, comparisons indicate that there is a very good agreement between the solutions of homotopy analysis method and the exact solutions in terms of accuracy.

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Fractional Whitham–Broer–Kaup Equations within Modified Analytical Approaches

Axioms ◽

10.3390/axioms8040125 ◽

2019 ◽

Vol 8 (4) ◽

pp. 125 ◽

Cited By ~ 7

Author(s):

Rasool Shah ◽

Hassan Khan ◽

Dumitru Baleanu

Keyword(s):

Fractional Order ◽

Traveling Wave ◽

Fractional Derivatives ◽

Traveling Wave Solution ◽

Order System ◽

Transform Method ◽

Homotopy Analysis ◽

Definition Of ◽

Analytical Approaches ◽

Natural Decomposition

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are compared with each other as well as with the exact results of the problems. The comparison shows the best agreement of solutions with each other and with the exact solution as well. Moreover, the proposed methods are found to be accurate, effective, and straightforward while dealing with the fractional-order system of partial differential equations and therefore can be generalized to other fractional order complex problems from engineering and science.

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Detecting Optimal Leak Locations Using Homotopy Analysis Method for Isothermal Hydrogen-Natural Gas Mixture in an Inclined Pipeline

Symmetry ◽

10.3390/sym12111769 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1769

Author(s):

Sarkhosh S. Chaharborj ◽

Zuhaila Ismail ◽

Norsarahaida Amin

Keyword(s):

Natural Gas ◽

Homotopy Analysis Method ◽

Numerical Solutions ◽

Gas Mixture ◽

Analysis Method ◽

Convergence Region ◽

Reduced Order ◽

Homotopy Analysis ◽

And Control ◽

Good Agreement

The aim of this article is to use the Homotopy Analysis Method (HAM) to pinpoint the optimal location of leakage in an inclined pipeline containing hydrogen-natural gas mixture by obtaining quick and accurate analytical solutions for nonlinear transportation equations. The homotopy analysis method utilizes a simple and powerful technique to adjust and control the convergence region of the infinite series solution using auxiliary parameters. The auxiliary parameters provide a convenient way of controlling the convergent region of series solutions. Numerical solutions obtained by HAM indicate that the approach is highly accurate, computationally very attractive and easy to implement. The solutions obtained with HAM have been shown to be in good agreement with those obtained using the method of characteristics (MOC) and the reduced order modelling (ROM) technique.

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Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense

Revista Mexicana de Física ◽

10.31349/revmexfis.64.539 ◽

2018 ◽

Vol 64 (5) ◽

pp. 539 ◽

Cited By ~ 14

Author(s):

Francisco Gomez ◽

Khaled Saad

Keyword(s):

Fractional Derivatives ◽

Error Function ◽

Reaction Diffusion ◽

Approximate Solutions ◽

Chemical System ◽

Transform Method ◽

Diffusion Waves ◽

Homotopy Analysis ◽

Optimal Values ◽

Coupled Reaction

In this paper, we have generalized the fractional cubic isothermal auto-catalytic chemical system (FCIACS) with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional time derivatives, respectively. We apply the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of FCIACS using these fractional derivatives. We study the convergence analysis of HATM by computing the residual error function. Also, we find the optimal values of h so we assure the convergence of the approximate solutions. Finally we show the behavior of the approximate solutions graphically. The results obtained are very effectiveness and accuracy.

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Approximate Analytic Solution for the KdV and Burger Equations with the Homotopy Analysis Method

Journal of Applied Mathematics ◽

10.1155/2012/878349 ◽

2012 ◽

Vol 2012 ◽

pp. 1-13 ◽

Cited By ~ 4

Author(s):

Mojtaba Nazari ◽

Faisal Salah ◽

Zainal Abdul Aziz ◽

Merbakhsh Nilashi

Keyword(s):

Homotopy Analysis Method ◽

Analytic Solution ◽

Order Approximation ◽

Soliton Solutions ◽

Nonlinear Problems ◽

Approximate Analytic Solution ◽

Analysis Method ◽

Convergence Region ◽

Burgers Equations ◽

Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate analytic solution of the Korteweg-de Vries (KdV) and Burgers equations. The homotopy analysis method (HAM) is an analytic technique which provides us with a new way to obtain series solutions of such nonlinear problems. HAM contains the auxiliary parameterħ, which provides us with a straightforward way to adjust and control the convergence region of the series solution. The resulted HAM solution at 8th-order and 14th-order approximation is then compared with that of the exact soliton solutions of KdV and Burgers equations, respectively, and shown to be in excellent agreement.

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