On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximations.

*(English)*Zbl 1413.65401Summary: In this article, multivariate Padé approximation and variational iteration method proposed by He is adopted for solving linear and nonlinear fractional partial differential equations. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include nonlinear time-fractional hyperbolic equation and linear fractional Klein-Gordon equation are investigated to show efficiency of multivariate Padé approximation. Comparison of the results obtained by the variational iteration method with those obtained by multivariate Padé approximation reveals that the present methods are very effective and convenient.

##### MSC:

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

41A21 | Padé approximation |

35R11 | Fractional partial differential equations |

35L70 | Second-order nonlinear hyperbolic equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

##### Keywords:

variational iteration method; multivariate Padé approximation; fractional differential equation; Caputo fractional derivative
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\textit{V. Turut} and \textit{N. Güzel}, Eur. J. Pure Appl. Math. 6, No. 2, 147--171 (2013; Zbl 1413.65401)

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##### References:

[1] | J. Abouir, A. Cuyt, P. Gonzalez-Vera, and R. Orive. On the Convergence of General Order Multivariate Padé-Type Approximants. Journal of Approximation Theory, 86:216–228, 1996. · Zbl 0860.41019 |

[2] | G. Adomian. A review of the decomposition method in applied mathematics. Fractional Calculus and Applied Analysis, 135:501–544, 1988. · Zbl 0671.34053 |

[3] | G. Adomian. Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston, 1994. · Zbl 0802.65122 |

[4] | M. Caputo. Linear models of dissipation whose Q is almost frequency independent. part II. Journal of the Royal Astronomical Society, 13:529–539, 1967. |

[5] | A. Cuyt. Multivariate Padé-approximant. Journal of Mathematical Analysis and Applica- tions, 96:283–293, 1983. |

[6] | A. Cuyt. A review of multivariate Padé approximation theory. Journal of Computational and Applied Mathematics, 12:221–232, 1985. · Zbl 0572.41010 |

[7] | A. Cuyt. How well can the concept of Padé approximant be generalized to the multivariate case? Journal of Computational and Applied Mathematics, 105:25–50, 1985. · Zbl 0945.41012 |

[8] | A. Cuyt and L. Wuytack. Nonlinear Methods in Numerical Analysis. Elsevier Science Publishers B.V, Amsterdam, 1987. · Zbl 0609.65001 |

[9] | A. Cuyt, L. Wuytack, and H. Werner. On the continuity of the multivariate Padé operator. Journal of Computational and Applied Mathematics, 11:95–102, 1984. · Zbl 0543.41018 |

[10] | L. Debnath and D. Bhatta. Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics. Fractional Calculus and Applied Analysis, 7:153– 192, 2004. |

[11] | R. Gorenflo. Afterthoughts on interpretation of fractional derivatives and integrals. In P. Rusev, I. Di-movski, and V. Kiryakovai, editors, Transform Methods and Special Func- tions, pages 589–591, Sofia, 1998. Bulgarian Academy of Sciences, Institute of Mathematics ands Informatics. REFERENCES169 |

[12] | Ph. Guillaume and A. Huard. Multivariate Padé Approximants. Journal of Computational and Applied Mathematics, 121:197–219, 2000. · Zbl 1090.41505 |

[13] | Ph. Guillaume, A. Huard, and V. Robin. Generalized Multivariate Padé Approximants. Journal of Approximation Theory, 95:203–214, 1998. · Zbl 0916.41015 |

[14] | J.H. He. Semi-inverse method of establishing generalized principlies for fluid mechanics with emphasis on turbomachinery aerodynamics. International Journal of Turbo and Jet Engines, 14(1):23–28, 1997. |

[15] | J.H. He. Variational iteration method for delay differential equations. Communications in Nonlinear Science and Numerical Simulation, 2(4):235–236, 1997. |

[16] | J.H. He. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering, 167:57–68, 1998. · Zbl 0942.76077 |

[17] | J.H. He.Approximate solution of nonlinear differential equations with convolution product nonlinearities. Computer Methods in Applied Mechanics and Engineering, 167:69–73, 1998. · Zbl 0932.65143 |

[18] | J.H. He. Nonlinear oscillation with fractional derivative and its applications. Interna- tional Conference on Vibrating Engineering ’98, pages 288–291, 1998. |

[19] | J.H. He. Some applications of nonlinear fractional differential equations and their approximations. Bulletin of Science and Technology, 15(2):86–90, 1999. |

[20] | J.H. He. Variational iteration method – a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics, 34:699–708, 1999. · Zbl 1342.34005 |

[21] | J.H. He. Variational iteration method for autonomous ordinary differential systems. Applied Mathematics and Computation, 114:115–123, 2000. · Zbl 1027.34009 |

[22] | J.H. He. Variational theory for linear magneto-electro-elasticity. International Journal of Nonlinear Sciences and Numerical Simulation, 2(4):309–316, 2001. |

[23] | J.H. He. Variational principle for Nano thin film lubrication. International Journal of Nonlinear Sciences and Numerical Simulation, 4(3):313–314, 2003. · Zbl 06942026 |

[24] | J.H. He. Variational principle for some nonlinear partial differential equations with variable coefficients. Chaos Solitons and Fractals, 19(4):847–851, 2004. · Zbl 1135.35303 |

[25] | J.H. He. Variational iteration method: Some recent results and new interpretations. Journal of Computational and Applied Mathematics, 207(1):3–17, 2007. · Zbl 1119.65049 |

[26] | J.H. He and X.H. Wu. Variational iteration method: New development and applications. Computers and Mathematics with Applications, 54(7-8):881–894, 2007. REFERENCES170 · Zbl 1141.65372 |

[27] | M. Inokuti, H. Sekine, and T. Mura. General use of the lagrange multiplier in non-linear mathematical physics. In S. Nemat-Nasser, editor, Variational Method in the Mechanics of Solids, pages 156–162, Oxford, 1978. Pergamon Press. |

[28] | A. Luchko and R. Groneflo. The initial value problem for some fractional differential equations with the Caputo derivative. Preprint series A0–98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1997. |

[29] | F. Mainardi. Fractional calculus: Some basic problems in continuum and statistical me- chanics. Springer-Verlag, New York, 1997. |

[30] | K.S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differ- ential Equations. John Wiley and Sons, Inc, New York, 1993. |

[31] | S. Momani. An explicit and numerical solutions of the fractional KdV equation. Mathe- matics and Computers in Simulation, 70(2):110–118, 2005. · Zbl 1119.65394 |

[32] | S. Momani. Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos, Solitons and Fractals, 28(4):930–937, 2006. · Zbl 1099.35118 |

[33] | S. Momani and S. Abuasad.Application of He’s variational iteration method to Helmholtz equation. Chaos Solitons and Fractals, 27(5):1119–1123, 2006. · Zbl 1086.65113 |

[34] | S. Momani and Z. Odibat. Analytical approach to linear fractional partial differential equations arising in fluild mechanics. Physics Letters A, 355:271–279, 2006. · Zbl 1378.76084 |

[35] | S. Momani and Z. Odibat.Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method. Applied Mathematics and Computation, 177:488–494, 2006. · Zbl 1096.65131 |

[36] | S. Momani and Z. Odibat. Approximate solutions for boundary value problems of timefractional wave equation. Applied Mathematics and Computation, 181:767–774, 2006. · Zbl 1148.65100 |

[37] | S. Momani and Z. Odibati. Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons and Fractals, 31:1248–1255, 2007. · Zbl 1137.65450 |

[38] | S. Momani and R. Qaralleh. Numerical approximations and Padé approximants for a fractional population growth model. Applied Mathematical Modelling, 31:1907–1914, 2007. · Zbl 1167.45300 |

[39] | S. Momani and N. Shawagfeh. Decomposition method for solving fractional Riccatti differential equations. Applied Mathematics and Computation, 182:1083–1092, 2006. · Zbl 1107.65121 |

[40] | Z. Odibat and S. Momani. An explicit and numerical solutions of the fractional KdV equation. International Journal of Nonlinear Sciences and Numerical Simulation, 7(1):15– 27, 2006. REFERENCES171 · Zbl 1401.65087 |

[41] | Z. Odibat and S. Momani. Numerical methods for nonlinear differential equations of fractional order. Applied Mathematical Modelling, 32:28–39, 2008. · Zbl 1133.65116 |

[42] | Z. Odibat and S. Momani. The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics. Computers and Mathematics with Applications, 58:2199–2208, 2009. · Zbl 1189.65254 |

[43] | K.B. Oldham and J. Spanier. The Fractional Calculus. Academic Press, New York, 1974. · Zbl 0292.26011 |

[44] | I. Podlubny. Fractional Differential Equations. Academic Press, New York, 1999. · Zbl 0924.34008 |

[45] | I. Podlubny. Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis, 5:367–386, 2002. · Zbl 1042.26003 |

[46] | A. Rèpaci. Nonlinear dynamical systems: On the accuracy of Adomian’s decomposition method. Applied Mathematics Letters, 3(3):35–39, 1990. · Zbl 0719.93041 |

[47] | V. Turut, E. Celik, and M. Yigider. Multivariate Padé approximation for solving partial differential equations (PDE). International Journal For Numerical Methods In Fluids, 66(9):1159–1173, 2011. |

[48] | A. Wazwaz. A new algorithm for calculating Adomian polynomials for nonlinear operators. Applied Mathematics and Computation, 111:53–69, 2000. · Zbl 1023.65108 |

[49] | A. Wazwaz and S. El-Sayed. A new modification of the Adomian decomposition method for linear and nonlinear operators. Applied Mathematics and Computation, 122:393–405, 2001. · Zbl 1027.35008 |

[50] | P. Zhou. Explicit construction of multivariate Padé approximants. Journal of Computa- tional and Applied Mathematics, 79:1–17, 1997. · Zbl 0865.41018 |

[51] | P. Zhou. Multivariate Padé Approximants Associated with Functional Relations. Journal of Approximation Theory, 93:201–230, 1998. · Zbl 0903.41007 |

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