Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations

In this paper, modified variational iteration algorithm-II is investigated for finding approximate solutions of nonlinear Parabolic equations. Comparisons of the MVIA-II with trigonometric B-spline collocation method, variational iteration method, homotopy perturbation transform method, Adomian decomposition method, and modified variational iteration method are carried out, which show that the proposed algorithm (MVIA-II) is robust one. Some nonlinear Parabolic equations are given to demonstrate the implementation and accuracy of the MVIA-II.

[1]  Md. Nur Alam.  Exact solutions to the foam drainage equation by using the new generalized (G′/G)-expansion method , 2015 .

[2]  M. Zellal,et al.  Applications of Homotopy Perturbation Transform Method for Solving Newell-Whitehead-Segel Equation , 2017 .

[3]  Junseok Kim,et al.  An explicit hybrid finite difference scheme for the Allen-Cahn equation , 2018, J. Comput. Appl. Math..

[4]  H. Kheiri,et al.  HOMOTOPY ANALYSIS AND HOMOTOPY PADE METHODS FOR THE MODIFIED BURGERS-KORTEWEG-DE VRIES AND THE NEWELL -WHITEHEAD EQUATIONS , 2011 .

[5]  Asıf Yokuş,et al.  On the numerical investigations to the Cahn-Allen equation by using finite difference method , 2018, An International Journal of Optimization and Control: Theories & Applications (IJOCTA).

[6]  Aly R. Seadawy,et al.  New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod , 2018 .

[7]  Md. Nur Alam,et al.  An analytical method for solving exact solutions of a nonlinear evolution equation describing the dynamics of ionic currents along microtubules , 2017 .

[8]  K. Kannan,et al.  Haar wavelet method for solving some nonlinear Parabolic equations , 2010, Forum for Interdisciplinary Mathematics.

[9]  Henry C. Tuckwell,et al.  Introduction to theoretical neurobiology , 1988 .

[10]  Gurhan Gurarslan.  Numerical modelling of linear and nonlinear diffusion equations by compact finite difference method , 2010 .

[11]  Aly R. Seadawy,et al.  Numerical solution of Korteweg–de Vries-Burgers equation by the modified variational iteration algorithm-II arising in shallow water waves , 2020, Physica Scripta.

[12]  João M. Nóbrega,et al.  Very high‐order accurate finite volume scheme for the convection‐diffusion equation with general boundary conditions on arbitrary curved boundaries , 2018, International Journal for Numerical Methods in Engineering.

[13]  Aly R. Seadawy,et al.  Mathematical methods via the nonlinear two-dimensional water waves of Olver dynamical equation and its exact solitary wave solutions , 2018 .

[14]  A. Seadawy.  Solitary wave solutions of two-dimensional nonlinear Kadomtsev–Petviashvili dynamic equation in dust-acoustic plasmas , 2017 .

[16]  Hijaz Ahmad,et al.  Numerical Solutions of Coupled Burgers′ Equations , 2019, Axioms.

[17]  Hijaz Ahmad,et al.  Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients , 2019, Comput. Math. Appl..

[18]  T. Tang,et al.  Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation , 2013 .

[19]  Hijaz Ahmad,et al.  Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations , 2019 .

[20]  Abdul-Majid Wazwaz,et al.  The Hirota's bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equation , 2008, Appl. Math. Comput..

[21]  Rongpei Zhang,et al.  The local discontinuous Galerkin method for Burger's-Huxley and Burger's-Fisher equations , 2012, Appl. Math. Comput..

[22]  F. Austin,et al.  THE VARIATIONAL ITERATION METHOD WHICH SHOULD BE FOLLOWED , 2010 .

[23]  W. Zahra.  Trigonometric B-Spline Collocation Method for Solving PHI-Four and Allen–Cahn Equations , 2017 .

[24]  T. Browder,et al.  Right-handed currents at $$B\rightarrow {K^{*}}\ell ^+\ell ^{-}$$B→K∗ℓ+ℓ- kinematic endpoint , 2017 .

[25]  Zigzag Instability and Reorientation of Roll Pattern in the Newell-Whitehead Equation , 1991 .

[26]  Minqiang Xu,et al.  An efficient reproducing kernel method for solving the Allen-Cahn equation , 2019, Appl. Math. Lett..

[27]  D. Lu,et al.  Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability , 2017 .

[28]  Abdul-Majid Wazwaz,et al.  The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations , 2007, Appl. Math. Comput..

[29]  Yaoyao Chen,et al.  Recovery type a posteriori error estimation of adaptive finite element method for Allen-Cahn equation , 2020, J. Comput. Appl. Math..

[30]  Gokul Hariharan,et al.  Haar wavelet method for solving Fisher's equation , 2009, Appl. Math. Comput..

[31]  Ji-Huan He,et al.  Lagrange crisis and generalized variational principle for 3D unsteady flow , 2019, International Journal of Numerical Methods for Heat & Fluid Flow.

[32]  Md. Nur Alam,et al.  An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator–prey system , 2016 .

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

[34]  Amit Prakash,et al.  HE'S VARIATIONAL ITERATION METHOD FOR THE SOLUTION OF NONLINEAR NEWELL-WHITEHEAD-SEGEL EQUATION , 2016 .

[35]  Aly R. Seadawy,et al.  Three-Dimensional Weakly Nonlinear Shallow Water Waves Regime and its Traveling Wave Solutions , 2017 .

[36]  Mostafa Safdari Shadloo,et al.  Numerical simulation of compressible flows by lattice Boltzmann method , 2019, Numerical Heat Transfer, Part A: Applications.

[37]  M. Moghimi,et al.  Variational iteration method for solving generalized Burger–Fisher and Burger equations , 2007 .

[38]  Luming Zhang,et al.  Error estimates of exponential wave integrator Fourier pseudospectral methods for the nonlinear Schrödinger equation , 2019, Appl. Math. Comput..

[39]  Hassan N. A. Ismail,et al.  Adomian decomposition method for Burger's-Huxley and Burger's-Fisher equations , 2004, Appl. Math. Comput..

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

[41]  Aly R. Seadawy,et al.  Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in unmagnetized dust plasma , 2018 .

[42]  Harun-Or Roshid,et al.  Exact traveling wave solutions to the (3+1)-dimensional mKdV–ZK and the (2+1)-dimensional Burgers equations via exp(−Φ(η))-expansion method , 2015 .

[43]  A. Patra,et al.  Homotopy perturbation sumudu transform method for solving convective radial fins with temperature-dependent thermal conductivity of fractional order energy balance equation , 2014 .

[44]  Nur Alam,et al.  Exact traveling wave solutions to higher order nonlinear equations , 2019, Journal of Ocean Engineering and Science.

[45]  Ranjith Maniyeri,et al.  Finite difference method based analysis of bio-heat transfer in human breast cyst , 2019, Thermal Science and Engineering Progress.

[46]  S. Antoniou.  The Riccati equation method with variable expansion coefficients. III. Solving the Newell-Whitehead equation , 2015 .

[47]  D. Frank-Kamenetskii,et al.  Diffusion and heat exchange in chemical kinetics , 1955 .

[48]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[49]  Abdul-Majid Wazwaz,et al.  An analytic study of Fisher's equation by using Adomian decomposition method , 2004, Appl. Math. Comput..

[50]  H. M. Baskonus,et al.  Some novel exponential function structures to the Cahn–Allen equation , 2016 .

[51]  Ji-Huan He,et al.  A variational principle for a thin film equation , 2019, Journal of Mathematical Chemistry.

[52]  Changfeng Gui,et al.  Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian , 2015 .

[53]  John J. Tyson,et al.  On Traveling Wave Solutions of Fisher's Equation in Two Spatial Dimensions , 1999, SIAM J. Appl. Math..

[54]  Reza Ezzati,et al.  Using adomian's decomposition and multiquadric quasi-interpolation methods for solving Newell-Whitehead equation , 2011, WCIT.

[56]  G. Hariharan,et al.  An Efficient Legendre Wavelet-Based Approximation Method for a Few Newell–Whitehead and Allen–Cahn Equations , 2014, The Journal of Membrane Biology.

[57]  E. Zayed,et al.  A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines , 2015 .

[58]  M. Javidi.  Modified pseudospectral method for generalized Burger's-Fisher equation , 2006 .

[59]  Ji-Huan He.  Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves , 2020 .

[60]  Aly R. Seadawy,et al.  Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity , 2019, Journal of Taibah University for Science.

[61]  Abdul-Majid Wazwaz,et al.  Trial equation method for solving the generalized Fisher equation with variable coefficients , 2016 .

[62]  E. Babolian,et al.  Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations , 2009 .

[63]  Ji-Huan He.  A modified Li-He’s variational principle for plasma , 2021, International Journal of Numerical Methods for Heat & Fluid Flow.