A note on variation iteration method with an application on Lane–Emden equations

PurposeIn this article, the authors consider the following nonlinear singular boundary value problem (SBVP) known as Lane–Emden equations, −u″(t)-(α/t) u′(t) = g(t, u), 0 < t < 1 where α ≥ 1 subject to two-point and three-point boundary conditions. The authors propose to develop a novel method to solve the class of Lane–Emden equations.Design/methodology/approachThe authors improve the modified variation iteration method (VIM) proposed in [JAAC, 9(4) 1242–1260 (2019)], which greatly accelerates the convergence and reduces the computational task.FindingsThe findings revealed that either exact or highly accurate approximate solutions of Lane–Emden equations can be computed with the proposed method.Originality/valueNovel modification is made in the VIM that provides either exact or highly accurate approximate solutions of Lane-Emden equations, which does not exist in the literature.

[1]  J. Keller,et al.  Electrohydrodynamics I. the Equilibrium of a Charged Gas in a Container , 2013 .

[2]  A. Verma,et al.  Haar wavelets collocation method for a system of nonlinear singular differential equations , 2020 .

[3]  Magdy A. El-Tawil,et al.  Solving nonlinear partial differential equations using the modified variational iteration Padé technique , 2007 .

[4]  M. M. Chawla,et al.  A Uniform Mesh Finite Difference Method for a Class of Singular Two-Point Boundary Value Problems , 1985 .

[5]  Magdy A. El-Tawil,et al.  Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Padé technique , 2007, Comput. Math. Appl..

[6]  A. Verma,et al.  Nonlinear three point Singular BVPs : A Classification , 2015, 1508.07408.

[7]  Zhaosheng Feng,et al.  Positive solutions of fractional differential equations with derivative terms , 2012 .

[8]  A. S. V. Ravi Kanth,et al.  Cubic spline for a class of non-linear singular boundary value problems arising in physiology , 2006, Appl. Math. Comput..

[9]  The monotone iterative method and zeros of Bessel functions for nonlinear singular derivative depend , 2011 .

[10]  G. Adomian,et al.  Inversion of nonlinear stochastic operators , 1983 .

[11]  S. H. Lin,et al.  Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. , 1976, Journal of theoretical biology.

[12]  Jitendra Kumar,et al.  The optimal modified variational iteration method for the Lane-Emden equations with Neumann and Robin boundary conditions , 2017 .

[13]  Swati,et al.  Higher order Emden-Fowler type equations via uniform Haar Wavelet resolution technique , 2020, J. Comput. Appl. Math..

[14]  Abdelhalim Ebaid,et al.  A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method , 2011, J. Comput. Appl. Math..

[15]  G. Adomian,et al.  Modified decomposition solution of linear and nonlinear boundary-value problems , 1994 .

[16]  Arvind K. Singh,et al.  On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology , 2004 .

[17]  Muhammad Aslam Noor,et al.  Modified variational iteration technique for solving singular fourth-order parabolic partial differential equations , 2009 .

[18]  Mandeep Singh,et al.  An effective computational technique for a class of Lane–Emden equations , 2015, Journal of Mathematical Chemistry.

[19]  L. Ahmad Soltani,et al.  A new modification of the variational iteration method , 2010, Comput. Math. Appl..

[20]  T. Schunck.  Zur Knickfestigkeit schwach gekrümmter zylindrischer Schalen , 1933 .

[21]  H. Sekine,et al.  General Use of the Lagrange Multiplier in Nonlinear Mathematical Physics1 , 1980 .

[22]  Lawrence F. Shampine,et al.  Numerical Methods for Singular Boundary Value Problems , 1975 .

[23]  Habibolla Latifizadeh,et al.  A general numerical algorithm for nonlinear differential equations by the variational iteration method , 2020 .

[24]  Ji-Huan He.  Variational iteration method—Some recent results and new interpretations , 2007 .

[25]  Abdul-Majid Wazwaz,et al.  The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models , 2011 .

[26]  R C Duggan,et al.  Pointwise bounds for a nonlinear heat conduction model of the human head. , 1986, Bulletin of mathematical biology.

[27]  Abdul-Majid Wazwaz,et al.  An algorithm based on the variational iteration technique for the Bratu-type and the Lane–Emden problems , 2016, Journal of Mathematical Chemistry.

[28]  Surendra Kumar,et al.  A note on the solution of singular boundary value problems arising in engineering and applied sciences: Use of OHAM , 2012, Comput. Chem. Eng..

[29]  Amit K. Verma,et al.  Higher resolution methods based on quasilinearization and Haar wavelets on Lane-Emden equations , 2019, Int. J. Wavelets Multiresolution Inf. Process..

[30]  R. Agarwal,et al.  ON AN ITERATIVE METHOD FOR A CLASS OF 2 POINT & 3 POINT NONLINEAR SBVPS , 2019, Journal of Applied Analysis & Computation.

[31]  A. Ghorbani,et al.  An effective modification of He’s variational iteration method , 2009 .

[32]  Junfeng Lu,et al.  Variational iteration method for solving two-point boundary value problems , 2007 .

[33]  Maximum and anti-maximum principles for three point SBVPs and nonlinear three point SBVPs , 2015 .

[34]  P. Chambré.  On the Solution of the Poisson‐Boltzmann Equation with Application to the Theory of Thermal Explosions , 1952 .

[35]  Suheil A. Khuri,et al.  A Laplace variational iteration strategy for the solution of differential equations , 2012, Appl. Math. Lett..

[36]  V. F. Kirichenko,et al.  Substantiation of the variational iteration method in the theory of plates , 1981 .

[37]  A. Verma.  MONOTONE ITERATIVE METHOD AND REGULAR SINGULAR NONLINEAR BVP IN THE PRESENCE OF REVERSE ORDERED UPPER AND LOWER SOLUTIONS , 2012 .

[38]  On a Constructive Approach for Derivative-Dependent Singular Boundary Value Problems , 2011 .

[39]  G. Adomian,et al.  A new algorithm for matching boundary conditions in decomposition solutions , 1993 .

[40]  R. K. Pandey.  A finite difference method for a class Of singular two point boundary value problems arising in physiology , 1997, Int. J. Comput. Math..

[41]  A. S. V. Ravi Kanth,et al.  He's variational iteration method for treating nonlinear singular boundary value problems , 2010, Comput. Math. Appl..

[42]  Ji-Huan He,et al.  Variational iteration method: New development and applications , 2007, Comput. Math. Appl..

[43]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[44]  R. Agarwal,et al.  A Review on a Class of Second Order Nonlinear Singular BVPs , 2020, Mathematics.

[45]  M. Chawla,et al.  A new spline method for singular two-point boundary value problems , 1988 .