[Paper Review] Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method

Pietro Artale Harris, Roberto Garra|arXiv (Cornell University)|Jun 8, 2013

Fractional Differential Equations Solutions8 references22 citations

TL;DR

This paper presents an analytic solution method for nonlinear time-fractional Burgers-type equations using the Invariant Subspace Method, enabling exact solutions for equations involving Caputo time-fractional derivatives. By identifying invariant subspaces spanned by power functions and special functions, the authors derive exact solutions through fractional differential systems, including forced and variable-coefficient variants, and extend the method to equations with both space- and time-fractional derivatives.

ABSTRACT

In this paper we study the analytic solutions of Burgers-type nonlinear fractional equations by means of the Invariant Subspace Method. We first study a class of nonlinear equations directly related to the time-fractional Burgers equation. Some generalizations linked to the forced time-fractional Burgers equations and variable-coefficient diffusion are also considered. Finally we study a Burgers-type equation involving both space and time-fractional derivatives.

Motivation & Objective

• To develop an exact analytic method for solving nonlinear time-fractional Burgers-type equations.
• To extend the Invariant Subspace Method to fractional differential equations with Caputo derivatives.
• To address the lack of exact solution techniques for nonlinear fractional PDEs in fluid dynamics and viscoelasticity.
• To generalize the method to forced equations and variable-coefficient diffusion terms.
• To apply the method to equations with both space- and time-fractional derivatives.

Proposed method

• The Invariant Subspace Method is applied to a nonlinear time-fractional diffusion equation related to the Burgers equation via differentiation with respect to x.
• The method identifies invariant subspaces spanned by functions such as $1$ and $x^\beta$, allowing solutions of the form $u(x,t) = a(t)x^\beta + \text{const}$.
• Exact solutions are derived by reducing the PDE to a system of fractional ODEs for the time-dependent coefficients $a(t), b(t), c(t)$.
• Fractional derivatives are treated in the Caputo sense, enabling the use of Riemann-Liouville integration and known fractional calculus identities.
• Solutions are constructed using fractional integrals and the properties of the Gamma function, particularly $J^\alpha t^{\delta}$ and $D^\alpha t^{\delta}$.
• The method is extended to equations with time-dependent diffusion coefficients $k(t)$ and to equations involving both space- and time-fractional derivatives.

Experimental results

Research questions

• RQ1 Can the Invariant Subspace Method yield exact analytic solutions for nonlinear time-fractional Burgers-type equations?
• RQ2 How can the method be adapted to equations with variable coefficients or external forcing terms?
• RQ3 What are the exact solutions for Burgers-type equations involving both space- and time-fractional Caputo derivatives?
• RQ4 How do the solutions behave in the limit $\alpha \to 1$ and $k \to 0$, and what is their relation to the classical Burgers equation?
• RQ5 What conditions ensure the existence and uniqueness of solutions in the fractional setting?

Key findings

• Exact solutions are obtained for the time-fractional Burgers equation by transforming it into a system of fractional ODEs via the invariant subspace approach.
• Solutions for the forced time-fractional Burgers equation are derived by solving a system of fractional ODEs involving $a(t), b(t), c(t)$, with $a(t)$ expressed via fractional integrals.
• For the equation with time-dependent diffusion coefficient $k(t)$, the solution exists as long as $k(t) \in L_1([0,t])$, and is constructed via fractional integration.
• An exact solution is found for the equation $\partial_t^\alpha u + u \partial_x^\beta u = 0$ in the form $u(x,t) = -\frac{\Gamma(1-\alpha)}{\Gamma(\beta+1)\Gamma(1-2\alpha)} \frac{x^\beta}{t^\alpha} + \text{const}$, valid for $\alpha \in (0,1) \setminus \{1/2\}$.
• The solution reduces to the classical Burgers equation solution in the limit $\alpha \to 1$, with $t$ replaced by $t^\alpha / C_\alpha$, where $C_\alpha = -\Gamma(1-\alpha)/\Gamma(1-2\alpha)$.
• The method is shown to be extendable to more complex fractional equations, with invariant subspaces generated by special functions of fractional calculus.