Abstract
The nonclassical symmetries method is a powerful extension of the classical symmetries method for finding exact solutions of differential equations. Through this method, one is able to arrive at new exact solutions of a given differential equation, i.e., solutions that are not obtainable directly as invariant solutions from classical symmetries of the equation. The challenge with the nonclassical symmetries method, however, is that governing equations for the admitted nonclassical symmetries are typically coupled and nonlinear and therefore difficult to solve. In instances where a given equation is related to a simpler one via an equivalent transformation, we propose that nonclassical symmetries of the given equation may be obtained by transforming nonclassical symmetries of the simpler equation using the equivalence transformation. This is what we illustrate in this paper. We construct four nontrivial nonclassical symmetries of the Black–Scholes equation by transforming nonclassical symmetries of the heat equation. For completeness, we also construct invariant solutions of the Black–Scholes equation associated with the determined nonclassical symmetries.
Highlights
E nonclassical symmetries method is a powerful extension of the classical symmetries method for finding exact solutions of differential equations. rough this method, one is able to arrive at new exact solutions of a given differential equation, i.e., solutions that are not obtainable directly as invariant solutions from classical symmetries of the equation. e challenge with the nonclassical symmetries method, is that governing equations for the admitted nonclassical symmetries are typically coupled and nonlinear and difficult to solve
In instances where a given equation is related to a simpler one via an equivalent transformation, we propose that nonclassical symmetries of the given equation may be obtained by transforming nonclassical symmetries of the simpler equation using the equivalence transformation. is is what we illustrate in this paper
Introduction ere is a lot of interest in the search for nonclassical symmetries of differential equations. is is because nonclassical symmetries lead to new solutions of differential equations, solutions not obtainable directly from classical Lie point symmetries of the equation
Summary
It is well known (see [17, 18, 24,25,26,27] and the references therein) that if a mapping from a given differential equation to another (target) differential equation is an invertible point transformation, the mapping establishes a one-to-one correspondence between symmetries of the given and target equations. Such mappings are realisable as equivalence transformations between equations that have similar symmetry Lie algebras. It turns out that nonclassical symmetries are among the properties that are mapped back and forth between the heat and Black–Scholes equations
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