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Methods of Solving PDEs

Table of Contents

A recent review (arXiv:2102.04815) noted:

There are several basic methods for finding exact solutions and constructing reductions of nonlinear partial differential equations: the method of group analysis of differential equations (the method of searching for classical symmetries) [1–6], methods for finding for non-classical symmetries [7–10], the direct Clarkson–Kruskal method [11–14], methods for generalized separation of variables [13–15], methods for functional separation of variables [14,16–18], the method of differential constraints [13,14,19], the method of truncated Painleve expansions [13,20,21], and use of conservation laws to obtain exact solutions [22–24].

1. Group Symmetry Methods

1.1. Classical Symmetries

  • Ovsiannikov, L.V.
    Group Analysis of Differential Equations.
    Academic Press, 1982.
  • GW Bluman, JW Cole,
    Similarity Methods for Differential Equations.
    Springer, 1974.
  • PJ Olver,
    Applications of Lie Groups to Differential Equations.
    2nd ed., Springer, 2000
  • VK Andreev, OV Kaptsov, VV Pukhnachov, AA Rodionov,
    Applications of Group-Theoretical Methods in Hydrodynamics.
    Kluwer, 1998.

1.2. Nonclassical Symmetry Reduction

  • Peter A. Clarkson, Elizabeth L. Mansfield,
    "Algorithms for the Nonclassical Method of Symmetry Reductions".
    arXiv:solv-int/9401002, 27 pages
  • Peter J Olver and Evgenii M Vorob'ev,
    "Nonclassical and Conditional Symmetries".
    In: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, N.H. Ibragimov, ed., CRC Press, Boca Raton, Fl., 1996, pp.291–328. PDF.

1.3. Approximate Symmetries

Also note there is another technique using "approximate symmetries", where the differential equation we are interested in solving may be viewed as a perturbation of a simpler differential equation. We may find the symmetries of the simpler equation, then use it as the basis for a perturbed symmetry group (approximate symmetry group) of our original differential equation.

  • Baikov, Gazizov, and Ibragimov,
    "Approximate symmetries".
    Math. USSR Sbomik 46 (1989) pp.427–441.
    • First publication inventing the technique
  • V. A. Baikov, R. K. Gazizov, N. Kh. Ibragimov,
    "Approximate symmetries".
    Mat. Sb. (N.S.), 136(178):4(8) (1988), 435–450; Math. USSR-Sb., 64:2 (1989), 427–441. Eprint

2. Differential Constraints

CRC Handbook of Nonlinear PDEs describe this method as: "The main idea of the method is that exact solutions to a complex (nonintegrable) equation are sought by jointly analyzing this equation and an auxiliary simpler (integrable) equation, called a differential constraint."

3. Clarkson-Kruskal method

  • P A Clarkson and M D Kruskal,
    "New similarity solutions of the Boussinesq equation".
    Journal of Mathematical Physics 30 (1989) pp 2201-2213.
    • First paper introducing the technique.
  • SHEN Shou-Feng,
    "Clarkson–Kruskal Direct Similarity Approach for Differential-Difference Equations".
    Commun. Theor. Phys. (Beijing, China) 44 (2005) pp. 964–966
  • AD Polyanin,
    "Comparison of the effectiveness of different methods for constructing exact solutions to nonlinear PDEs".
    Mathematics 7, no. 5 (2019) doi:10.3390/math7050386

4. Functional Separation of Variables

The basic idea is to solve the problem

\begin{equation} F(x, u_{x}, u_{t}, u_{xx}, u_{xt}, u_{tt}, \dots) = 0 \end{equation}

The separation of variables usually takes the form \(u(x,t) = X(x)T(t)\), but the functional separation of variables uses the implicit form

\begin{equation} \int h(u)\,\D u = \xi(x)\omega(t) + \eta(x). \end{equation}

Where in Heaven's name does this come from? And what on Earth is \(h(u)\), \(\xi(x)\), \(\omega(t)\), and \(\eta(x)\)?

Well, this is a generalization of the nonlinear heat equation

\begin{equation} u_{t} = \partial_{x}[f(u)u_{x}] \end{equation}

where \(f(u)\) is an arbitrary function. This admits a traveling-wave solution \(u=u(z)\) for \(z=\lambda t + \kappa x\), which can be represented in the implicit form

\begin{equation} \kappa^{2}\int\frac{f(u)}{\lambda u + C_{1}}\D u = \lambda t + \kappa x + C_{2} \end{equation}

The generalization is carried out as follows:

\[ \kappa^{2}\frac{f(u)}{\lambda u + C_{1}} \to h(u) \]

\[ \lambda\to \xi(x) \]

\[ t \to \omega(t) \]

\[ \kappa x + C_{2}\to \eta(x).\]

  • Andrei D. Polyanin,
    "Construction of exact solutions in implicit form for PDEs: New functional separable solutions of non-linear reaction-diffusion equations with variable coefficients".
    May 2019 International Journal of Non-Linear Mechanics 111:95-105 DOI: 10.1016/j.ijnonlinmec.2019.02.005, Eprint
  • Andrei D. Polyanin,
    "Functional separation of variables in nonlinear PDEs: General approach, new solutions of diffusion-type equations".
    arXiv:2001.01645, 56 pages

Last Updated 2021-06-01 Tue 10:00.