“…Nucci and Clarkson [71] (see also [21]) demonstrated that the symmetry reduction (5.14) of the Fitzhugh-Nagumo equation (1.9), is obtainable using the nonclassical method due to Bluman and Cole [9], though not using the direct method due to Clarkson and Kruskal [20]. Hence the nonclassical method is more general than the direct method, at least as it was originally formulated.…”
Section: Discussionmentioning
confidence: 99%
“…The condition that 27d + 2b 3 − 9bc = 0 is the necessary and sufficient condition on f (u) = u 3 + bu 2 + cu + d for the existence of nonclassical symmetry reductions of (1.1) to odes solvable in terms of elliptic functions. Furthermore, Nucci and Clarkson [71] show that elliptic functions solutions exist for the Fitzhugh-Nagumo equation (1.9) if either a = −1, a = 1 2 or a = 2, which are precisely those the three cases when the distances from the outer roots to the central one are equal.…”
Section: Nonclassical Symmetries τ ≡mentioning
confidence: 96%
“…Travelling wave solutions of the Fitzhugh-Nagumo equation (1.9) have been studied by several authors [6,7,28,42,96]. Exact solutions of (1.9) have been obtained using various techniques including Vorob'ev [97] (who calls the associated symmetry a "partial symmetry of the first type"), by Kawahara and Tanaka [55] using Hirota's bi-linear method [50], by Hereman [45] using the truncated Painlevé expansion method (see also [17]) and more recently by Nucci and Clarkson [71] using the nonclassical method (see also [21]). Exact solutions of the Huxley equation…”
Section: ) Is Invariant Under This Transformation Yields An Overdetementioning
confidence: 99%
“…Following Nucci and Clarkson [71], this is solvable as follow. Using (5.5) to eliminate u t in (5.1) yields …”
Classical and nonclassical symmetries of the nonlinear heat equationare considered. The method of differential Gröbner bases is used both to find the conditions on f (u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f (u) in terms of the roots of f (u) = 0. 0
“…Nucci and Clarkson [71] (see also [21]) demonstrated that the symmetry reduction (5.14) of the Fitzhugh-Nagumo equation (1.9), is obtainable using the nonclassical method due to Bluman and Cole [9], though not using the direct method due to Clarkson and Kruskal [20]. Hence the nonclassical method is more general than the direct method, at least as it was originally formulated.…”
Section: Discussionmentioning
confidence: 99%
“…The condition that 27d + 2b 3 − 9bc = 0 is the necessary and sufficient condition on f (u) = u 3 + bu 2 + cu + d for the existence of nonclassical symmetry reductions of (1.1) to odes solvable in terms of elliptic functions. Furthermore, Nucci and Clarkson [71] show that elliptic functions solutions exist for the Fitzhugh-Nagumo equation (1.9) if either a = −1, a = 1 2 or a = 2, which are precisely those the three cases when the distances from the outer roots to the central one are equal.…”
Section: Nonclassical Symmetries τ ≡mentioning
confidence: 96%
“…Travelling wave solutions of the Fitzhugh-Nagumo equation (1.9) have been studied by several authors [6,7,28,42,96]. Exact solutions of (1.9) have been obtained using various techniques including Vorob'ev [97] (who calls the associated symmetry a "partial symmetry of the first type"), by Kawahara and Tanaka [55] using Hirota's bi-linear method [50], by Hereman [45] using the truncated Painlevé expansion method (see also [17]) and more recently by Nucci and Clarkson [71] using the nonclassical method (see also [21]). Exact solutions of the Huxley equation…”
Section: ) Is Invariant Under This Transformation Yields An Overdetementioning
confidence: 99%
“…Following Nucci and Clarkson [71], this is solvable as follow. Using (5.5) to eliminate u t in (5.1) yields …”
Classical and nonclassical symmetries of the nonlinear heat equationare considered. The method of differential Gröbner bases is used both to find the conditions on f (u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f (u) in terms of the roots of f (u) = 0. 0
“…The exact solution (40) (see Ref. [22]) can also be obtained with a nonclassical method of symmetry reduction [23] or via Hirota bilinear method. [24] …”
Recently, the authors of [Commun. Theor. Phys. 56 (2011) 397] made a number of useful observations on Exp-function method. In this study, we focus on another vital issue, namely, the misleading results of double Exp-function method.
In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Fitzhugh-Nagumo equation. Numerical solutions obtained by these methods when compared with the exact solutions reveal that the obtained solutions produce high accurate results. The results show that the HPM, the VIM and the ADM are of high accuracy and are efficient for solving the Fitzhugh-Nagumo equation. Also the results demonstrate that the introduced methods are powerful tools for solving the nonlinear partial differential equations.
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