Approximate and Analytic Solution of Some Nonlinear Diffusive Equations

  • Amitha Manmohan Rao
  • Arundhati Suresh Warke
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 41)


Nonlinear partial differential equations (PDEs) have wide range of applications in mathematics, science and engineering and are used in modelling various types of problems arising in fluid mechanics. This paper presents the numerical approximation of some nonlinear diffusive PDEs; Newell–Whitehead–Segel (NWS) equation and Burgers’ equation by using Laplace decomposition method (LDM) and finite difference method(FDM). The nonlinear PDEs are considered to study the influence of the parameters like initial condition and dissipative coefficient on the solution, wave distortion, and wave propagation. The numerical results obtained are analysed graphically. The approach can be extended to obtain physically relevant solutions to a wide range of nonlinear PDEs describing various real life phenomena involving nonlinear and dissipative effects. MATLAB-R2017b is used for all the computations and graphical representation.


Adomian polynomials Approximate solution FDM Nonlinear PDE NWS equation LDM 


  1. 1.
    J.M. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordtrecht, 1974.CrossRefGoogle Scholar
  2. 2.
    V.I. Karpman, Non-Linear Waves in Dispersive Media, Pergamon, Oxford, 1975.CrossRefGoogle Scholar
  3. 3.
    S.A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics. 1(2001), 141-155.MathSciNetCrossRefGoogle Scholar
  4. 4.
    S.A. Khuri, A new approach to Bratus problem, Applied Mathematics and Computation. 1(2004), 131-136.MathSciNetCrossRefGoogle Scholar
  5. 5.
    H. Jafari, Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations, Applied Mathematics Letters. 24(2011), 1799-1805.MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. A. Hussain, Modified Laplace Decomposition Method, Applied Mathematical Science. 4(2010), 1769-1783.Google Scholar
  7. 7.
    M. E. Khan, Application of Laplace Decomposition Method to Solve Nonlinear Coupled Partial Differential Equations, World Applied Sciences Journal, 8(2010), 13-19.Google Scholar
  8. 8.
    Y. A. Khan, Application of modified Laplace decomposition method for solving boundary layer equation, Journal of King Saud University-Science, 23(2011), 115-119.CrossRefGoogle Scholar
  9. 9.
    J. Fadaei, Application of Laplace−Adomian Decomposition Method on Linear and Nonlinear System of PDEs, Applied Mathematical Sciences, 5(2011), 1307-1315.Google Scholar
  10. 10.
    J. Patad, S. Bhalekar, Approximate analytical solutions of Newell-Whitehead-Segel equation using a new iterative method, World Journal of Modelling and Simulation. 11(2015), 94-103.Google Scholar
  11. 11.
    M. M. A. Mahgoub and A.K.H. Sedeeg, On The Solution of Newell-Whitehead-Segel Equation, American Journal of Mathematical and Computer Modelling. 1(2016), 21-24.
  12. 12.
    P. Pue-on, Laplace Adomian Decomposition Method for Solving Newell-Whitehead-Segel Equation, Applied Mathematical Sciences. 7(2013), 6593 – 6600.MathSciNetCrossRefGoogle Scholar
  13. 13.
    M. Hussain and M. Khan, Modified Laplace Decomposition Method, Applied Mathematical Sciences. 4(2010), 1769 - 1783Google Scholar
  14. 14.
    A. M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied Mathematics and Computation. 111(2000), 33-5.MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. M. Rao, and A. S. Warke, Laplace Decomposition Method (LDM) for Solving Nonlinear Gas Dynamic Equation, Annals of the Faculty of Engineering Hunedoara-International Journal of Engineering. 2(2015), 147-150.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Amitha Manmohan Rao
    • 1
    • 2
  • Arundhati Suresh Warke
    • 3
  1. 1.Symbiosis International (Deemed University)PuneIndia
  2. 2.N.S.S. College of Commerce & EconomicsAffiliated to University of MumbaiMumbaiIndia
  3. 3.Symbiosis Institute of TechnologyPuneIndia

Personalised recommendations