Group formalism of Lie transformations, conservation laws, exact and numerical solutions of non-linear time-fractional Black–Scholes equation

Numerical conservation laws of time fractional diffusion PDEs

“…Other promising approaches for the numerical solution of time fractional PDEs are pseudo spectral methods based on fractional Lagrange and Müntz-Legendre polynomials [31,32,47,54]. A future development of this research could deal with the analysis of the conservative properties of these classes of methods and their experimental verification to make comparisons with the proposed approach.…”

“…For nonlinearly self-adjoint FDEs that do not have a Lagrangian in the classical sense, a formal Lagrangian can be introduced and conservation laws are obtained by using modern techniques based on Lie group analysis of FDEs. This approach, proposed for the first time by Lukashchuk in 2015 [41], has been applied to time fractional PDEs [1, 12, 29, 36, 38-41, 47, 51] and more recently to time and space fractional PDEs (see [54,56] and references therein).…”

“…Within this class, we recall methods based on Chelyshkov polynomials [46,48] and on Chebyshev polynomials [49,50]. Pseudo-spectral methods based on fractional Lagrange or Müntz-Legendre polynomials have been introduced for the solution of time fractional PDEs, such as Fokker-Plank equation [31], Klein-Gordon equation [47], Black-Scholes equation [54], and a diffusion equation [32]. The rest of the paper is organized as follows.…”

On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems

“…In view of Theorem 3 one can conclude that any symmetry of this system should have the form The solution of this system with respect to the functions ψ(x), θ(x), ϕ 1 (x), ϕ 2 (x), η 1 (0) (t, x) and η 2 (0) (t, x) leads to the following result. The system (40) with arbitrary σ = 0 and β = 0 has four linearly independent infinitesimal symmetries…”

“…At present, numerous symmetries, invariant solutions, and conservation laws have been obtained for wide classes of FDEs describing various anomalous processes and phenomena (see, e.g., [32][33][34][35][36][37][38][39][40][41] and references therein). Nevertheless, finding symmetries of FDEs is a more complex problem than that for integer-order differential equations.…”