Lie symmetries reduction and spectral methods on the fractional two-dimensional heat equation

Abstract: In this paper, the Lie symmetry analysis is proposed for a space-time convection-diffusion fractional differential equations with the Riemann-Liouville derivative by (2+1) independent variables and one dependent variable. We find a reduction form of our governed fractional differential equation using the similarity solution of our Lie symmetry. One-dimensional optimal system of Lie symmetry algebras is found. We present a computational method via the spectral method based on Bernstein's operational matrices to solve the two-dimensional fractional heat equation with some initial conditions.

• The paper employs Lie symmetry analysis to reduce a space-time fractional 2D heat equation to a simpler form via similarity solutions.
• It introduces a spectral method using Bernstein polynomials to convert fractional derivatives into manageable algebraic equations.
• Numerical examples validate the approach, demonstrating convergence and accuracy in approximating solutions for fractional orders.

Lie Symmetries and Spectral Methods for the Fractional Two-Dimensional Heat Equation

This paper addresses the problem of finding exact solutions and numerical approximations for a space-time fractional convection-diffusion equation with two spatial dimensions and one temporal dimension. The authors employ Lie symmetry analysis to reduce the governing fractional differential equation and then apply a spectral method based on Bernstein polynomials to obtain numerical solutions.

Lie Symmetry Analysis

The paper begins by performing a Lie symmetry analysis on the (2+1)-dimensional fractional convection-diffusion equation:

$$D^\alpha_t u(t, x, y)=D^\beta_x u(t, x, y)+D^\beta_y u(t, x, y)+f(x, y,t),$$

where $0<\alpha\leq1,~~0<\beta<1$ and $D^\alpha_t$, $D^\beta_x$ and $D^\beta_x$ are the Riemann-Liouville fractional derivatives of order $\alpha$ and $\beta$ with respect to the variable $t, x$ and $y$ and $f$ is an arbitrary smooth function.

The Lie symmetry algebra is found to be spanned by the following vector fields:

$${\bf X}_1={\partial\over\partial t},~~~ {\bf X}_2={\partial\over\partial x},~~~ {\bf X}_3={\partial\over\partial y},~~~ {\bf X}_4=u{\partial\over\partial u},$$

$${\bf X}_5=\alpha t{\partial\over\partial t}+\beta x{\partial\over\partial x}+\beta y{\partial\over\partial y},~~~{\bf X}_6=g(t, x, y){\partial\over\partial u}.$$

The authors then construct a one-dimensional optimal system of Lie symmetry algebras to classify the possible symmetry reductions. This involves finding a set of subalgebras that are non-conjugate under the adjoint action of the Lie group.

Symmetry Reduction and Similarity Solutions

Using the obtained Lie symmetries, the authors perform a symmetry reduction of the fractional differential equation. Specifically, they utilize the similarity solution derived from the vector field ${\bf X}_5$:

$u=\omega(\xi_1, \xi_2), \;\;\;\; \xi_1=xt^{-\frac{\beta}{\alpha},\;\; \xi_2=yt^{-\frac{\beta}{\alpha},$

to transform the original equation into a reduced equation involving the extended left-hand sided Erdelyi-Kober fractional derivative operator.

Spectral Method Based on Bernstein Polynomials

To obtain numerical solutions, the authors employ a spectral method based on Bernstein polynomials. This involves approximating the solution using Bernstein basis functions and then using operational matrices to represent the fractional derivatives. The key idea is to leverage the properties of Bernstein polynomials to transform the fractional differential equation into a system of algebraic equations, which can then be solved numerically.

The method is applied to the two-dimensional fractional heat equation with specific initial conditions. The results are compared with the exact solution for the case when $\alpha=1$ and $\beta=2$, demonstrating good agreement. The authors also present approximate solutions for fractional values of $\alpha$ and $\beta$, showing the behavior of the solution as the fractional order approaches the integer order.

Numerical Example

The paper presents a numerical example to illustrate the application of the spectral method. The two-dimensional fractional heat conduction equation is given by:

$$D^\alpha_t u(t, x, y)=D^\beta_x u(t, x, y)+D^\beta_y u(t, x, y)+2tx^3y^3-6t^2xy^3-6t^2x^3y,$$

where $0<\alpha \leq 1,\; 1<\beta \leq 2$ and $t, x,y\in [0,1]$. The exact solution for $\alpha= 1$ and $\beta= 2$ is $u(t, x, y)=t^2x^3y^3$.

The absolute error between the approximate and exact solutions is calculated at different time and space points, showing good agreement. The approximate solutions for fractional values of $\alpha$ with $\beta =2$ are also presented, demonstrating the convergence of the approximate solutions to the exact solution as $\alpha$ approaches 1.

Conclusion

The paper presents a combination of Lie symmetry analysis and spectral methods for solving the fractional two-dimensional heat equation. The Lie symmetry analysis is used to reduce the complexity of the problem, while the spectral method provides an efficient way to obtain numerical solutions. The numerical example demonstrates the accuracy and effectiveness of the proposed method.