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Research Papers

A Robust Computational Algorithm of Homotopy Asymptotic Method for Solving Systems of Fractional Differential Equations

[+] Author and Article Information
Zaid Odibat

Department of Mathematics,
Faculty of Science,
Al-Balqa Applied University,
Salt 19117, Jordan;
School of Basic Sciences and Humanities,
German Jordanian University,
Amman 11180, Jordan
e-mails: odibat@bau.edu.jo;
z.odibat@gmail.com

Sunil Kumar

Department of Mathematics,
National Institute of Technology,
Jamshedpur, Jharkhand 801014, India

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 31, 2018; final manuscript received April 22, 2019; published online May 13, 2019. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 14(8), 081004 (May 13, 2019) (10 pages) Paper No: CND-18-1489; doi: 10.1115/1.4043617 History: Received October 31, 2018; Revised April 22, 2019

In this paper, we present new ideas for the implementation of homotopy asymptotic method (HAM) to solve systems of nonlinear fractional differential equations (FDEs). An effective computational algorithm, which is based on Taylor series approximations of the nonlinear equations, is introduced to accelerate the convergence of series solutions. The proposed algorithm suggests a new optimal construction of the homotopy that reduces the computational complexity and improves the performance of the method. Some numerical examples are tested to validate and illustrate the efficiency of the proposed algorithm. The obtained results demonstrate the improvement of the accuracy by the new algorithm.

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References

Liao, S. , 1997, “ Homotopy Analysis Method: A New Analytical Technique for Nonlinear Problems,” Commun. Nonlinear Sci. Numer. Simul., 2(2), pp. 95–100.
Liao, S. , 2003, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, FL.
Liao, S. , 2004, “ On the Homotopy Analysis Method for Nonlinear Problems,” Appl. Math. Comput., 147, pp. 499–513.
Liao, S. , and Tan, Y. , 2007, “ A General Approach to Obtain Series Solutions of Nonlinear Differential Equations,” Stud. Appl. Math., 119(4), pp. 297–354.
Liao, S. , 2009, “ Notes on the Homotopy Analysis Method: Some Definitions and Theorems,” Commun. Nonlinear Sci. Numer. Simul., 14(4), pp. 983–997.
Wang, Z. , Zou, L. , and Zhang, H. , 2007, “ Applying Homotopy Analysis Method for Solving Differential-Difference Equation,” Phys. Lett. A, 369(1–2), pp. 77–84.
Abbasbandy, S. , 2008, “ Approximate Solution for the Nonlinear Model of Diffusion and Reaction in Porous Catalysts by Means of the Homotopy Analysis Method,” Chem. Eng. J., 136(2–3), pp. 144–150.
Molabahrami, A. , and Khani, F. , 2009, “ The Homotopy Analysis Method to Solve the Burgers-Huxley Equation,” Nonlinear Anal.: Real World Appl., 10(2), pp. 589–600.
Rashidi, M. , and Dinarvand, S. , 2009, “ Purely Analytic Approximate Solutions for Steady Three-Dimensional Problem of Condensation Film on Inclined Rotating Disk by Homotopy Analysis Method,” Nonlinear Anal.: Real World Appl., 10(4), pp. 2346–2356.
Chen, Y. , and Liu, J. , 2009, “ A Study of Homotopy Analysis Method for Limit Cycle of Van Der Pol Equation,” Comm. Nonlinear Sci. Numer. Simul., 14(5), pp. 1816–1821.
Odibat, Z. , 2010, “ A Study on the Convergence of Homotopy Analysis Method,” Appl. Math. Comput., 217(2), pp. 782–789.
Martin, O. , 2013, “ On the Homotopy Analysis Method for Solving a Particle Transport Equation,” Appl. Math. Model., 37(6), pp. 3959–3967.
Shivanian, E. , and Abbasbandy, S. , 2014, “ Predictor Homotopy Analysis Method: Two Points Second Order Boundary Value Problems,” Nonlinear Anal.: Real World Appl., 15, pp. 89–99.
Kumar, S. , Singh, J. , Kumar, D. , and Kapoor, S. , 2014, “ New Homotopy Analysis Transform Algorithm to Solve Volterra Integral Equation,” Ain Shams Eng. J., 5(1), pp. 243–246.
Massa, F. , Lallemand, B. , and Tison, T. , 2015, “ Multi-Level Homotopy Perturbation and Projection Techniques for the Reanalysis of Quadratic Eigenvalue Problems: The Application of Stability Analysis,” Mech. Syst. Signal Process., 52, pp. 88–104.
Sardanys, J. , Rodrigues, C. , Janurio, C. , Martins, N. , Gil-Gmez, G. , and Duarte, J. , 2015, “ Activation of Effector Immune Cells Promotes Tumor Stochastic Extinction: A Homotopy Analysis Approach,” Appl. Math. Comput., 252(1), pp. 484–495.
Hetmaniok, E. , Słota, D. , Wituła, R. , and Zielonka, A. , 2015, “ Solution of the One-Phase Inverse Stefan Problem by Using the Homotopy Analysis Method,” Appl. Math. Model., 39(22), pp. 6793–6805.
Odibat, Z. , and Bataineh, A. , 2015, “ An Adaptation of HAM for Reliable Treatment of Strongly Nonlinear Problems: Construction of Homotopy Polynomials,” Math. Methods Appl. Sci., 38(5), pp. 991–1000.
Liu, Q. X. , Liu, J. K. , and Chen, Y. M. , 2016, “ Asymptotic Limit Cycle of Fractional Van Der Pol Oscillator by Homotopy Analysis Method and Memory-Free Principle,” Appl. Math. Model., 40(4), pp. 3211–3220.
Yang, Z. , and Liao, S. , 2017, “ A HAM-Based Wavelet Approach for Nonlinear Partial Differential Equations: Two Dimensional Bratu Problem as an Application,” Comm. Nonlinear Sci. Numer. Simul., 53, pp. 249–262.
Gorder, R. , and Vajravelu, K. , 2009, “ On the Selection of Auxiliary Functions, Operators, and Convergence Control Parameters in the Application of the Homotopy Analysis Method to Nonlinear Differential Equations: A General Approach,” Comm. Nonlinear Sci. Numer. Simul., 14(12), pp. 4078–4089.
Liao, S. , 2010, “ An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations,” Comm. Nonlinear Sci. Numer. Simul., 15(8), pp. 2003–2016.
Pandey, R. , Singh, O. , Baranwal, V. , and Tripathi, M. , 2012, “ An Analytic Solution for the Space-Time Fractional Advection-Dispersion Equation Using the Optimal Homotopy Asymptotic Method,” Comput. Phys. Commun., 183(10), pp. 2089–2106.
Golbabai, A. , Fardi, M. , and Sayevand, K. , 2013, “ Application of the Optimal Homotopy Asymptotic Method for Solving a Strongly Nonlinear Oscillatory System,” Math. Comput. Model., 58(11–12), pp. 1837–1843.
Marinca, V. , and Herisanu, N. , 2014, “ The Optimal Homotopy Asymptotic Method for Solving Blasius Equation,” Appl. Math. Comput., 231, pp. 134–139.
Mallory, K. , and Gorder, R. , 2014, “ Optimal Homotopy Analysis and Control of Error for Solutions to the Non-Local Whitham Equation,” Numer. Algorithms, 66(4), pp. 843–863.
Sarwar, S. , Alkhalaf, S. , Iqbal, S. , and Zahid, M. A. , 2015, “ A Note on Optimal Homotopy Asymptotic Method for the Solutions of Fractional Order Heat- and Wave-Like Partial Differential Equations,” Comput. Math. Appl., 70(5), pp. 942–953.
Hamarsheh, M. , Ismail, A. , and Odibat, Z. , 2015, “ Optimal Homotopy Asymptotic Method for Solving Fractional Relaxation-Oscillation Equation,” J. Interpolat. Approx. Sci. Comput., 2015(2), pp. 98–111.
Jia, W. , He, X. , and Guo, L. , 2017, “ The Optimal Homotopy Analysis Method for Solving Linear Optimal Control Problems,” Appl. Math. Model., 45, pp. 865–880.
Oldham, K. B. , and Spanier, J. , 1974, The Fractional Calculus, Academic Press, New York.
Miller, K. S. , and Ross, B. , 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Gorenflo, R. , and Mainardi, F. , 1997, “ Fractional Calculus: Integral and Differential Equations of Fractional Order,” Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi , eds., Springer Verlag, New York, pp. 277–290.
Hilfer, R. , 2000, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore.
Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands.
Lorenzo, C. F. , and Hartley, T. T. , 2016, The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, Wiley, New York.
Zaky, M. A. , Doha, E. H. , and Tenreiro Machado, J. A. , 2018, “ A Spectral Numerical Method for Solving Distributed-Order Fractional Initial Value Problems,” ASME J. Comput. Nonlinear Dyn., 13(10), p. 101007.
David, S. A. , Quintino, D. D. , Inacio, C. M. C. , and Machado, J. A. T. , 2018, “ Fractional Dynamic Behavior in Ethanol Prices Series,” J. Comput. Appl. Math., 339, pp. 85–93.
Jajarmi, A. , and Baleanu, D. , 2018, “ A New Fractional Analysis on the Interaction of HIV With CD4+ T-Cells,” Chaos Solitons Fractals, 113, pp. 221–229.
Baleanu, D. , Jajarmi, A. , Bonyah, E. , and Hajipour, M. , 2018, “ New Aspects of Poor Nutrition in the Life Cycle Within the Fractional Calculus,” Adv. Differ. Equations, 2018, p. 230.
Jajarmi, A. , and Baleanu, D. , 2018, “ Suboptimal Control of Fractional-Order Dynamic Systems With Delay Argument,” J. Vib. Control, 24(12), pp. 2430–2446.
Momani, S. , and Al-Khaled, K. , 2005, “ Numerical Solutions for Systems of Fractional Differential Equations by the Decomposition Method,” Appl. Math. Comput., 162(3), pp. 1351–1365.
Jafari, H. , and Daftardar-Gejji, V. , 2006, “ Revised Adomian Decomposition Method for Solving Systems of Ordinary and Fractional Differential Equations,” Appl. Math. Comput., 181(1), pp. 598–608.
Momani, S. , and Odibat, Z. , 2007, “ Numerical Approach to Differential Equations of Fractional Order,” J. Comput. Appl. Math., 207(1), pp. 96–110.
Jafari, H. , and Seifi, S. , 2009, “ Solving a System of Nonlinear Fractional Partial Differential Equations Using Homotopy Analysis Method,” Comm. Nonlinear Sci. Numer. Simul., 14(5), pp. 1962–1969.
Odibat, Z. M. , Corson, N. , Aziz-Alaoui, M. A. , and Bertelle, C. , 2010, “ Synchronization of Chaotic Fractional-Order Systems Via Linear Control,” Int. J. Bifurcation Chaos, 20(1), pp. 81–97.
Odibat, Z. , 2010, “ Analytic Study on Linear Systems of Fractional Differential Equations,” Comput. Math. Appl., 59(3), pp. 1171–1183.
Erturk, V. , Odibat, Z. , and Momani, S. , 2011, “ An Approximate Solution of a Fractional Order Differential Equation Model of Human T-Cell Lymphotropic Virus I (HTLV-I) Infection of CD4+ T-Cells,” Comput. Math. Appl., 62(3), pp. 996–1002.
Odibat, Z. , 2012, “ A Note on Phase Synchronization in Coupled Chaotic Fractional Order Systems,” Nonlinear Anal.: Real World Appl., 13(2), pp. 779–789.
Bhrawy, A. H. , and Zaky, M. A. , 2016, “ Shifted Fractional-Order Jacobi Orthogonal Functions: Application to a System of Fractional Differential Equations,” Appl. Math. Model., 40(2), pp. 832–845.
Ouannas, A. , Odibat, Z. , and Hayat, T. , 2017, “ Fractional Analysis of Co-Existence of Some Types of Chaos Synchronization,” Chaos Solitons Fractals, 105, pp. 215–223.
Ouannas, A. , Grassi, G. , Ziar, T. , and Odibat, Z. , 2017, “ On a Function Projective Synchronization Scheme for Non-Identical Fractional-Order Chaotic (Hyperchaotic) Systems With Different Dimensions and Orders,” Optik, 136, pp. 513–523.
Du, W. , Miao, Q. , Tong, L. , and Tang, Y. , 2017, “ Identification of Fractional-Order Systems With Unknown Initial Values and Structure,” Phys. Let. A, 381(23), pp. 1943–1949.
Shukla, M. K. , and Sharma, B. B. , 2017, “ Backstepping Based Stabilization and Synchronization of a Class of Fractional Order Chaotic Systems,” Chaos Solitons Fractals, 102, pp. 274–284.
Odibat, Z. , Corson, N. , Aziz-Alaoui, M. A. , and Alsaedi, A. , 2017, “ Chaos in Fractional Order Cubic Chua System and Synchronization,” Int. J. Bifurcation Chaos, 27(10), p. 1750161.
Wang, J. , Xu, T. Z. , Wei, Y. Q. , and Xie, J. Q. , 2018, “ Numerical Simulation for Coupled Systems of Nonlinear Fractional Order Integro-Differential Equations Via Wavelets Method,” Appl. Math. Comput., 324, pp. 36–50.
Lenka, B. K. , and Banerjee, S. , 2018, “ Sufficient Conditions for Asymptotic Stability and Stabilization of Autonomous Fractional Order Systems,” Comm. Nonlinear Sci. Numer. Simul., 56, pp. 365–379.
Cang, J. , Tan, Y. , Xu, H. , and Liao, S. , 2009, “ Series Solutions of Non-Linear Riccati Differential Equations With Fractional Order,” Chaos Solitons Fractals, 40(1), pp. 1–9.
Odibat, Z. , Momani, S. , and Xu, H. , 2010, “ A Reliable Algorithm of Homotopy Analysis Method for Solving Nonlinear Fractional Differential Equations,” Appl. Math. Model., 34(3), pp. 593–600.
Zurigat, M. , Momani, S. , Odibat, Z. , and Alawneh, A. , 2010, “ The Homotopy Analysis Method for Handling Systems of Fractional Differential Equations,” Appl. Math. Model., 34(1), pp. 24–35.
Odibat, Z. , 2011, “ On Legendre Polynomial Approximation With the VIM or HAM for Numerical Treatment of Nonlinear Fractional Differential Equations,” J. Comput. Appl. Math., 235(9), pp. 2956–2968.
Kumar, S. , Kumar, A. , and Odibat, Z. , 2017, “ A Nonlinear Fractional Model to Describe the Population Dynamics of Two Interacting Species,” Math. Methods Appl. Sci., 40(11), pp. 4134–4148.
Odibat, Z. , 2019, “ On the Optimal Selection of the Linear Operator and the Initial Approximation in the Application of the Homotopy Analysis Method to Nonlinear Fractional Differential Equations,” Appl. Numer. Math., 137, pp. 203–212.

Figures

Grahic Jump Location
Fig. 1

Plots of approximate solutions Σm=0Nxm(t) of x(t) and exact solution for the system given in Eq. (24), when α1 = α2 = 1 and N =10: (------) exact solution using RK4; (- - - - -) approximate solution using standard HAM; (−·−·) approximate solution using computational algorithm of HAM. (a) ℏ1=−1, ℏ2=−1, (b) ℏ1=−1.2, ℏ2=−1.1, (c) ℏ1=−1.25, ℏ2=−0.5, and (d) ℏ1=−0.8, ℏ2=−1.2.

Grahic Jump Location
Fig. 2

Plots of approximate solutions Σm=0Nxm(t) of x(t) for the system given in Eq. (24), when ℏ1=ℏ2=−1 and N =12: (−−−) approximate solution using standard HAM; (−·−·) approximate solution using computational algorithm of HAM. (a) α1 = α2 = 0.85 and (b) α1 = α2 = 0.75.

Grahic Jump Location
Fig. 3

Numerical results of ρi's for the system given in Eq. (24) on I = [0, 1], when ℏ1=ℏ2=−1, using computational algorithm of HAM. (a) α1 = α2 = 0.9 and (b) α1 = α2 = 0.75.

Grahic Jump Location
Fig. 4

Plots of approximate solutions Σm=0Nxm(t) and Σm=0Nym(t) of x(t) and y(t), respectively, and exact solution for the system given in Eq. (33), when α1=α2=1, N =10 and ℏ1=ℏ2=−1: (-----) exact solution using RK4; (−−−) approximate solution using standard HAM; (−·−·) approximate solution using computational algorithm of HAM. (a) a = 1, b = 0.5, δ = 1.5, γ = 2, (b) a = 2, b = 1.5, δ = –1, γ = 1, and (c) a = 2, b = 0.25, δ = 1, γ = 1.5.

Grahic Jump Location
Fig. 5

Plots of approximate solutions for the fractional Lotka–Volterra system given in Eq. (38), when a =0.5, b =0.25, c =1, d =0.5, δ = 1.3 and γ = 0.6, in the case of ℏ1=ℏ2=−1 and N =10: (-----) x(t)≈Σm=0Nxm(t); (−−−)y(t)≈Σm=0Nym(t). (a) α1 = α2 = 0.7, (b) α1 = α2 = 0.8, (c) α1 = α2 = 0.9, and (d) α1 = α2 = 1.

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