Laplace transform homotopy perturbation method for the approximation of variational problems

Abstract This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.

[1]  Randolph Rach,et al.  A bibliography of the theory and applications of the Adomian decomposition method, 1961-2011 , 2012 .

[2]  Hooman Fatoorehchi,et al.  Series solution of nonlinear differential equations by a novel extension of the Laplace transform method , 2016, Int. J. Comput. Math..

[3]  H. Fatoorehchi,et al.  A more realistic approach toward the differential equation governing the glass transition phenomenon , 2013 .

[4]  F. Casas.  Solution of linear partial differential equations by Lie algebraic methods , 1996 .

[5]  Y. Shang.  Lie algebra method for solving biological population model , 2013 .

[6]  Hossein Aminikhah,et al.  The combined Laplace transform and new homotopy perturbation methods for stiff systems of ODEs , 2012 .

[7]  H. Salamati,et al.  The effect of MgO doping on the structural, magnetic, and magnetotransport properties of La0.8Sr0.2MnO3 manganite , 2013 .

[8]  Jafar Biazar,et al.  On the order of convergence of Adomian method , 2002, Appl. Math. Comput..

[9]  Antonio Marin-Hernandez,et al.  Nonlinearities distribution Laplace transform-homotopy perturbation method , 2014, SpringerPlus.

[10]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

[11]  Arturo Sarmiento-Reyes,et al.  Analytical Solutions for Systems of Singular Partial Differential-Algebraic Equations , 2015 .

[12]  Janet Willars,et al.  The effect of vaginal closure technique on early post-operative pain following vaginal prolapse surgery: a feasibility pilot study and qualitative assessment , 2014, SpringerPlus.

[13]  Hooman Fatoorehchi,et al.  Investigation of Nonlinear Problems of Heat Conduction in Tapered Cooling Fins Via Symbolic Programming , 2012 .

[14]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[15]  Hooman Fatoorehchi,et al.  Chaos control in the cerium-catalyzed Belousov–Zhabotinsky reaction using recurrence quantification analysis measures , 2015 .

[16]  Ji-Huan He,et al.  Recent development of the homotopy perturbation method , 2008 .

[17]  R. Rach,et al.  An accurate explicit form of the Hankinson–Thomas–Phillips correlation for prediction of the natural gas compressibility factor , 2014 .

[18]  Mohsen Razzaghi,et al.  Legendre wavelets method for the solution of nonlinear problems in the calculus of variations , 2001 .

[19]  Hooman Fatoorehchi,et al.  Feedback control strategies for a cerium-catalyzed Belousov–Zhabotinsky chemical reaction system , 2015 .

[20]  Nicolae Herisanu,et al.  Nonlinear dynamical systems in engineering , 2011 .

[21]  Yasir Khan,et al.  USING PERTURBATION METHODS AND LAPLACE-PAD´ E APPROXIMATION TO SOLVE NONLINEAR PROBLEMS , 2013 .

[22]  S. Ghaderi.  Homotopy Perturbation Method for Solving Moving Boundary and Isoperimetric Problems , 2012 .

[23]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[24]  Gamal N. Elnagar,et al.  A pseudospectral collocation method for the brachistochrone problem , 1994 .

[25]  Mohammad Mehdi Rashidi,et al.  A STUDY OF NON-NEWTONIAN FLOW AND HEAT TRANSFER OVER A NON-ISOTHERMAL WEDGE USING THE HOMOTOPY ANALYSIS METHOD , 2012 .

[26]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[27]  Hossein Aminikhah,et al.  A novel effective approach for solving nonlinear heat transfer equations , 2012 .

[28]  U. Filobello-Niño,et al.  Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals , 2015 .

[29]  C. Lanczos.  The variational principles of mechanics , 1949 .

[30]  HeJi-Huan,et al.  Homotopy Perturbation Method , 2003, Advanced Numerical and Semi-Analytical Methods for Differential Equations.

[31]  H. Fatoorehchi,et al.  Analytical Solution to Intra-Phase Mass Transfer in Falling Film Contactors via Homotopy Perturbation Method , 2011 .

[32]  Michael R. Osborne,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[33]  Yasir Khan,et al.  Extension of Laplace transform–homotopy perturbation method to solve nonlinear differential equations with variable coefficients defined with Robin boundary conditions , 2017, Neural Computing and Applications.

[34]  George Adomian,et al.  On the solution of algebraic equations by the decomposition method , 1985 .

[35]  Hossein Aminikhah.  Analytical Approximation to the Solution of Nonlinear Blasius’ Viscous Flow Equation by LTNHPM , 2012 .

[36]  Yilun Shang,et al.  A LIE ALGEBRA APPROACH TO SUSCEPTIBLE-INFECTED-SUSCEPTIBLE EPIDEMICS , 2012 .

[37]  G. F. Simmons.  Differential Equations With Applications and Historical Notes , 1972 .

[38]  Hooman Fatoorehchi,et al.  Analytical approximate solutions for a general nonlinear resistor–nonlinear capacitor circuit model , 2015 .

[39]  Hooman Fatoorehchi,et al.  A novel and computationally efficient algorithm for stability analysis of multi input-multi output process control systems , 2015, Korean Journal of Chemical Engineering.

[40]  Alejandro Díaz-Sánchez,et al.  Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version , 2012, J. Appl. Math..

[41]  Héctor Vázquez-Leal,et al.  A generalization of the Bernoulli's method applied to brachistochrone-like problems , 2013, Appl. Math. Comput..

[42]  U. Filobello-Niño,et al.  Modified HPMs Inspired by Homotopy Continuation Methods , 2012 .

[43]  L. Elsgolts.  Differential Equations and the Calculus of Variations , 2003 .

[44]  Y. Shang,et al.  Analytical Solution for an In-host Viral Infection Model with Time-inhomogeneous Rates , 2015 .

[45]  G. Adomian.  A review of the decomposition method in applied mathematics , 1988 .

[46]  Ji-Huan He.  A coupling method of a homotopy technique and a perturbation technique for non-linear problems , 2000 .

[47]  Fei Xu.  A Generalized Soliton Solution of the Konopelchenko-Dubrovsky Equation using He’s Exp-Function Method , 2007 .

[48]  I. Hashim.  Adomian decomposition method for solving BVPs for fourth-order integro-differential equations , 2006 .

[49]  Mohammad Mehdi Rashidi,et al.  Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method , 2012 .

[50]  David J. Evans,et al.  The tanh function method for solving some important non-linear partial differential equations , 2005, Int. J. Comput. Math..

[51]  Arturo Sarmiento-Reyes,et al.  The study of heat transfer phenomena using PM for approximate solution with Dirichlet and mixed boundary conditions , 2013 .

[52]  Davood Domiri Ganji,et al.  Explicit Solution of Nonlinear ZK-BBM Wave Equation Using Exp-Function Method , 2008 .

[53]  U. Filobello,et al.  An Approximate Solution of Blasius Equation by using HPM Method , 2012 .

[54]  M. Lakestani,et al.  SOLVING BRACHISTOCHRONE PROBLEM USING HOMOTOPY ANALYSIS METHOD , 2013 .

[55]  Antonio Marin-Hernandez,et al.  A handy approximate solution for a squeezing flow between two infinite plates by using of Laplace transform-homotopy perturbation method , 2014, SpringerPlus.