Exact solution to a class of problems for the Burgers' equation on bounded intervals

Technical Summary: Exact Solution to a Class of Problems for the Burgers' Equation on Bounded Intervals

Problem Statement
The paper addresses the viscous Burgers' equation, a nonlinear partial differential equation modeling the balance between convection and diffusion, defined on generic bounded intervals $[l_1, l_2]$. The specific problem involves determining the function $w(x, t)$ subject to:

1. Governing Equation: $w_t - a^2 w_{xx} + w w_x = 0$, where $a^2$ represents the kinematic viscosity coefficient ($1/Re$).
2. Initial Condition: A non-homogeneous initial profile $w(x, 0) = w_0(x)$.
3. Boundary Conditions: Fixed Dirichlet boundary conditions $w(l_1, t) = \alpha_1$ and $w(l_2, t) = \alpha_2$.

The authors note that while numerical methods (finite difference, finite element, spectral methods) and semi-analytical techniques (Adomian decomposition, homotopy perturbation) are prevalent, they often lack closed-form solutions or require significant computational resources. The goal is to derive an exact operational solution that offers a closed-form expression in the Laplace domain and a precise time-domain solution via inverse transforms.

Methodology
The proposed solution strategy relies on a two-step transformation process:

1. Hopf-Cole Transformation: The nonlinear Burgers' equation is linearized using the classical Hopf-Cole transformation:
   $$w(x, t) = -2a^2 \frac{u_x(x, t)}{u(x, t)}$$
   This converts the nonlinear problem into a linear boundary value problem for the one-dimensional heat diffusion equation (reaction-diffusion with zero reaction term):
   $$u_t - a^2 u_{xx} = 0$$
   The initial condition for $u$ is derived from the integral of the initial condition $w_0$, and the Dirichlet conditions for $w$ transform into homogeneous Robin boundary conditions for $u$.

2. Laplace Transform and Operational Solution: The linear heat equation is solved in the Laplace domain ($p$-domain). The authors utilize a previously established exact operational solution for linear reaction-diffusion equations with constant coefficients on finite intervals.

   • The problem is formulated as a two-point boundary value problem in the Laplace domain.
   • An exact expression for the Laplace transform $U(x, p)$ is derived, involving exponential terms dependent on $\sqrt{p}$ and a linear system of four equations to determine boundary values ($U(l_1, p), U_x(l_1, p), U(l_2, p), U_x(l_2, p)$).
   • The source term in the linearized equation is zero, simplifying the operational solution to depend on the initial condition transform $R(x, p)$.

3. Time-Domain Recovery: The exact solution for the original variable $w(x, t)$ is obtained implicitly via the inverse Laplace transform:
   $$w(x, t) = -2a^2 \frac{\mathcal{L}^{-1}\{U_x(x, p)\}}{\mathcal{L}^{-1}\{U(x, p)\}}$$
   The paper notes that while analytic inverses may exist in closed form (potentially via Mellin transforms), highly efficient numerical algorithms (specifically the de Hoog et al. algorithm implemented as invlap.m in Matlab) are employed to compute the time-domain solution regardless of the complexity of the Laplace expressions.

Key Contributions and Results
The paper presents two illustrative examples to validate the methodology:

• Example 1: A problem with an initial condition $w_0(x) = \frac{2a^2\pi \sin(\pi x)}{\sigma + \cos(\pi x)}$ and zero boundary conditions. The authors derived the Laplace domain solution and applied the inverse transform. The resulting time-domain solution matched the known classical exact closed-form solution (Wood, 2014) perfectly.
• Example 2: A problem with an initial condition $w_0(x) = \sin(\pi x)$. The classical exact solution for this case is known to be an infinite series. The authors' method produced an implicit exact solution in the Laplace domain. When numerically inverted, the results aligned closely with the truncated infinite series solution ("Theoretical solution").

Comparative Performance
The study highlights two primary advantages of the proposed method over existing approaches:

1. Computational Efficiency: In Example 2, the numerical inverse Laplace procedure was found to be more than 100 times faster in execution time compared to the evaluation of the truncated infinite series solution.
2. Accuracy: The numerical inverse Laplace solutions showed negligible error compared to the theoretical series solutions. In contrast, finite difference solutions (computed via Matlab's pdepe()) showed increasing discrepancy as the viscosity coefficient decreased, particularly at early time steps.

Significance and Claims
The authors claim that their work provides a closed-form operational solution in the Laplace domain and an exact implicit solution in the time domain. They assert that this approach offers a "novel alternative" to solutions based on infinite series or purely numerical discretization.

The significance of the work is framed as follows:

• Reference Point: The derived exact solution serves as a robust benchmark for validating and testing other numerical and semi-analytical methods.
• Efficiency: The method is computationally superior to series expansions, which can be resource-intensive.
• Generality: The methodology handles smooth solutions as well as generalized solutions with discontinuities in derivatives.

The paper concludes modestly, stating that the solution methodology will be a reference for future work, including potential extensions to forced Burgers' equations in higher dimensions and comparisons with other nonlinear evolution equation techniques like Painlevé analysis and Hirota's bilinear technique.