An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: Construction of homotopy polynomials

Mathematical Methods in the Applied Sciences

Volumn 38, Issue 5, 2015, Pages 991-1000

  Odibat, Zaid ^a   Sami Bataineh, A ^a  

^a AL BALQA APPLIED UNIVERSITY   (Jordan)

Author keywords

homotopy analysis method; homotopy polynomials; nonlinear differential equation; Taylor series; the pendulum equation; the quadratic Riccati equation

Indexed keywords

DIFFERENTIAL EQUATIONS; NONLINEAR ANALYSIS; NONLINEAR EQUATIONS; NUMERICAL METHODS; PROBLEM SOLVING; RICCATI EQUATIONS; TAYLOR SERIES;

HOMOTOPIES; HOMOTOPY ANALYSIS METHODS; MATHEMATICAL PHYSICS; NONLINEAR DIFFERENTIAL EQUATION; NONLINEAR PROBLEMS; PENDULUM EQUATIONS; SOLUTION PROCEDURE; STRONGLY NONLINEAR;

POLYNOMIALS;

EID: 84988268365     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.3136     Document Type: Article

References (41)

1. Nayfeh A,. Perturbation Methods. John Wiley and Sons: New York, 2000.
2. Liao S,. A kind of approximate solution technique which does not depend upon small parameters: a special example. International Journal of Non-Linear Mechanics 1995; 30: 371-380.
3. Liao S,. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press: Boca Raton, 2003.
4. Liao S,. On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation 2004; 147: 499-513.
5. Liao S, Tan Y,. A general approach to obtain series solutions of nonlinear differential equations. Studies in Applied Mathematics 2007; 119: 297-354.
6. Liao S,. Notes on the homotopy analysis method: some definitions and theorems. Communications in Nonlinear Science and Numerical Simulation 2009; 14 (4): 983-997.
7. Hayat T, Khan M, Sajid M, Asghar S,. Rotating flow of a third grade fluid in a porous space with hall current. Nonlinear Dynamics 2007; 49: 83-91.
8. Hayat T, Sajid M,. Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid. International Journal of Engineering Science 2007; 45: 393-401.
9. Song L, Zhang H,. Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation. Physics Letters A 2007; 367 (1-2): 88-94.
10. Bataineh A, Noorani M, Hashim I,. Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method. Physics Letters A 2007; 371 (1-2): 72-82.
11. Wang Z, Zhu L, Zhang H,. Applying homotopy analysis method for solving differential-difference equation. Physics Letters A 2007; 369 (1-2): 77-84.
12. Abbasbandy S,. Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method. Chemical Engineering Journal 2008; 136 (2-3): 144-150.
13. Abbasbandy S, Magyari E, Shivanian E,. The homotopy analysis method for multiple solutions of nonlinear boundary value problems. Communications in Nonlinear Science and Numerical Simulation 2009; 14 (9-10): 3530-3536.
14. Molabahrami A, Khani F,. The homotopy analysis method to solve the Burgers-Huxley equation. Nonlinear Analysis: Real World Applications 2009; 10 (2): 589-600.
15. Rashidi M, Dinarvand S,. Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method. Nonlinear Analysis: Real World Applications 2009; 10 (4): 2346-2356.
16. Liao S,. Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method. Nonlinear Analysis: Real World Applications 2009; 10 (4): 2455-2470.
17. Hashim I, Abdulaziz O, Momani S,. Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation 2009; 14 (3): 674-684.
18. Cang J, Tan Y, Xu H, Liao S,. Series solutions of non-linear Riccati differential equations with fractional order. Chaos, Solitons & Fractals 2009; 40 (1): 1-9.
19. Rashidi M, Domairry G, Dinarvand S,. Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation 2009; 14 (3): 708-717.
20. Chen Y, Liu J,. A study of homotopy analysis method for limit cycle of van der Pol equation. Communications in Nonlinear Science and Numerical Simulation 2009; 14 (5): 1816-1821.
21. Jafari H, Seifi S,. Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation 2009; 14 (5): 2006-2012.
22. Alomari A, Noorani M, Nazar R,. Adaptation of homotopy analysis method for the numeric-analytic solution of Chen system. Communications in Nonlinear Science and Numerical Simulation 2009; 14 (5): 2336-2346.
23. Zurigat M, Momani S, Odibat Z, Alawneh A,. The homotopy analysis method for handling systems of fractional differential equations. Applied Mathematical Modelling 2010; 34 (1): 24-35.
24. Odibat Z, Momani S, Xu H,. A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations. Applied Mathematical Modelling 2010; 34 (3): 593-600.
25. Odibat Z,. A study on the convergence of homotopy analysis method. Applied Mathematics and Computation 2010; 217 (2): 782-789.
26. Odibat Z,. On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equations. Journal of Computational and Applied Mathematics 2011; 235 (9): 2956-2968.
27. Abbasbandy S, Shirzadi A,. A new application of the homotopy analysis method: Solving the Sturm-Liouville problems. Communications in Nonlinear Science and Numerical Simulation 2011; 16 (1): 112-126.
28. Wang Q,. The optimal homotopy-analysis method for Kawahara equation. Nonlinear Analysis: Real World Applications 2011; 12 (3): 1555-1561.
29. Zhao Y, Lin Z, Liu Z, Liao S,. The improved homotopy analysis method for the Thomas-Fermi equation. Applied Mathematics and Computation 2012; 218 (17): 8363-8369.
30. Hassan H, El-Tawil M,. A new technique of using homotopy analysis method for second order nonlinear differential equations. Applied Mathematics and Computation 2012; 219 (2): 708-728.
31. Shukla A, Ramamohan T, Srinivas S,. Homotopy analysis method with a non-homogeneous term in the auxiliary linear operator. Communications in Nonlinear Science and Numerical Simulation 2012; 17 (10): 3776-3787.
32. Tsai C,. Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations. Engineering Analysis with Boundary Elements 2012; 36 (8): 1226-1234.
33. Sedighi H, Shirazi K, Zare J,. An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method. International Journal of Non-Linear Mechanics 2012; 47 (7): 777-784.
34. Rashidi M, Pour S, Hayat T, Obaidat S,. Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method. Computers and Fluids 2012; 54: 1-9.
35. Guerrero F, Santonja F, Villanueva R,. Solving a model for the evolution of smoking habit in Spain with homotopy analysis method. Nonlinear Analysis: Real World Applications 2013; 14 (1): 549-558.
36. Martin O,. On the homotopy analysis method for solving a particle transport equation. Applied Mathematical Modelling 2013; 37 (6): 3959-3967.
37. Liu C,. The essence of the homotopy analysis method. Applied Mathematical Modelling 2010; 216 (4): 1299-1303.
38. Abbasbandy S,. Homotopy perturbation method for solving quadratic Riccati differential equation and comparison with Adomian's decomposition method. Applied Mathematics and Computation 2006; 172: 485-490.
39. Abbasbandy S,. Homotopy analysis method for solving quadratic Riccati differential equation. Communications in Nonlinear Science and Numerical Simulation 2008; 13: 539-546.
40. Duan J,. New recurrence algorithms for the nonclassic Adomian polynomials. Computers & Mathematcs with Applications 2011; 62: 2961-2977.
41. Bataineh A, Noorani M, Hashim I,. On a new reliable modification of homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation 2009; 14 (2): 409-423.