Abstract
So far in this text we have been mainly concerned in applying classic methods, the Adomina decomposition method [3–5], and the variational iteration method [8–10] in studying first order and second order linear partial differential equations. In this chapter, we will focus our study on the nonlinear partial differential equations. The nonlinear partial differential equations arise in a wide variety of physical problems such as fluid dynamics, plasma physics, solid mechanics and quantum field theory. Systems of nonlinear partial differential equations have been also noticed to arise in chemical and biological applications. The nonlinear wave equations and the solitons concept have introduced remarkable achievements in the field of applied sciences. The solutions obtained from nonlinear wave equations are different from the solutions of the linear wave equations [1–2].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, (1991).
M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, (1994).
G. Adominan, A new approach to nonlinear partial differential equations, J. Math. Anal. Appl., 102, 420–434, (1984).
G. Adominan, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986).
Y. Cherruault, Convergence of Adomian’s method, Kybernotes, 18(2), 31–38, (1989).
Y. Cherruault and G. Adominan, Decomposition methods: a new proof of convergence, Math. Comput. Modelling, 18, 103–106, (1993).
J.H. He, Some asymptotic methods for strongly nonlinearly equations, Int J. Modern Math., 20(10), 1141–1199, (2006).
J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. math. Comput., 114(2/3), 115–123, (2000).
J.H. He, Variational iteration method—a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech, 34, 799–708, (1999).
A. Répaci, Nonlinear dynamical systems: on the accuracy of Adominan’s decomposition method, Appl. Math. Lett., 3, 35–39, (1990).
P. Rosenau and J.M. Hyman, Compactons: Solitons with finite wavelengths, Phys. Rev. Lett., 70(5), 564–567, (1993).
A.M. Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput., 81, 199–204, (1997).
A.M. Wazwaz, A reliable modification of Adomian’s decomposition method, Appl. Math. and Comput., 92, 1–7, (1998).
A.M. Wazwaz, A new technique for calculating Adominan polynomials for nonlinear operators, Appl. Math. Comput., 111(1), 33–51, (2000).
A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Leiden, (2002).
A.M. Wazwaz, The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations, Comput. Math. Applic. 54, 926–932, (2007).
A.M. Wazwaz, The variational iteration method for solving linear and nonlinear systems of PDEs, Comput. Math. Applic. 54, 895–902, (2007).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2009 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wazwaz, AM. (2009). Nonlinear Partial Differential Equations. In: Partial Differential Equations and Solitary Waves Theory. Nonlinear Physical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00251-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-00251-9_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00250-2
Online ISBN: 978-3-642-00251-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)