Implicit Scheme Solves Fractional Burgers Equation Using Caputo-Fabrizio Non-Singular Derivative

A research team comprising mathematicians L. Azami, A. H. Refahi Sheikhani, and H. Saberi Najafi has formulated and theoretically validated an implicit numerical method using the Crank–Nicholson finite difference technique combined with the Caputo–Fabrizio fractional time derivative to solve the nonlinear fractional Burgers equation. Grounded in continuum mechanics and expressway traffic flow modeling, the mathematical formulation achieves unconditional numerical stability while delivering theoretical and computational convergence with first-order accuracy in time and second-order accuracy in space. The investigation addresses a foundational one-dimensional equation derived directly from the incompressible Navier–Stokes equations by dropping pressure gradient terms and setting external forcing to zero under constant fluid density. By establishing rigorous Fourier analysis proofs alongside exact analytical benchmark solutions, the authors demonstrate that their implicit numerical formulation circumvents the mathematical deficits historically associated with singular memory kernels while preserving the delicate physical balance between nonlinear convective advection and dissipative diffusion.

In fluid mechanics and applied mathematics, the one-dimensional Burgers equation serves as an essential nonlinear parabolic model derived directly from the incompressible Navier–Stokes momentum equations. Within the complete Navier–Stokes framework, fluid density rho, velocity u, pressure p, dynamic viscosity mu, and external force G dictate fluid transport. By omitting the spatial pressure gradient, the momentum balance simplifies into an intermediate nonlinear formulation. Assuming that the external force G equals zero and fluid density rho remains constant, the governing equation reduces to the classical viscid Burgers equation containing a temporal acceleration derivative, a nonlinear convective advection term u du/dx, and a viscous diffusion term nu d^2u/dx^2, where nu represents kinematic viscosity. When kinematic viscosity nu reaches zero, the relationship transitions into the inviscid Burgers equation, representing scalar conservation laws. Physical phenomena captured by this governing structure include acoustic transmission, fluid dynamics, shock wave development, gas dynamics, pipe flow, granular silo discharge, and airflow around aircraft wings.

Beyond traditional aerodynamic and fluid mechanical problems, the nonlinear Burgers equation provides a powerful theoretical bridge connecting fluid dynamics with expressway traffic flow theory. In traffic flow modeling based on the classical Greenshields formulation, vehicular concentration across a roadway is represented in a moving reference frame where vehicle concentration relates linearly to drift speed in one spatial dimension. Expressway traffic current is defined quantitatively as the number of vehicles passing a designated reference observation point per unit of time. Under these modeling assumptions, the Burgers equation directly tracks vehicle concentration profiles as they evolve along the expressway. The authors note that a primary motivation for their study is to observe how diffusion and nonlinear advection reach equilibrium, particularly examining whether and how the mathematical limit of small kinematic viscosity nu aligns with scalar conservation law theory. In this small-viscosity regime, where diffusion is weak and nonlinear convective effects dominate, solutions converge toward entropy solutions.

To enhance mathematical modeling accuracy, fractional calculus extends traditional differential and integral operators to non-integer fractional orders alpha, offering powerful mathematical tools for describing dynamical systems with hereditary properties and memory effects, such as viscoelastic materials and anomalous diffusion. Although fractional derivative studies date back to seventeenth-century investigations by Leibniz and Newton, traditional fractional operators—including the Grunwald–Letnikov, Riemann–Liouville, and classical Caputo derivatives—are defined using singular power-law kernels. Because these singular kernels contain an intrinsic singularity at the origin, they cannot completely depict full memory effects or multiscale structural heterogeneities. To overcome this limitation, Caputo and Fabrizio introduced an alternative definition of the fractional time derivative that replaces the singular power-law kernel with an exponential non-singular kernel and substitutes the gamma function normalization with a normalization factor M(alpha) satisfying M(0) = M(1) = 1. Defined on Sobolev space H^1(a,b) or extended to L^1(-infinity, b), this non-singular formulation provides robust multiscale structural modeling.

To discretize the fractional Burgers equation subject to prescribed spatial boundary conditions and initial conditions, the authors construct a uniform computational grid across the spatial interval [0, X] and time domain [0, T]. Spatial discretization is established with grid spacing h = Delta x = 1/M across spatial node indices i ranging from 0 to M, while temporal discretization employs uniform step size k = Delta t = T/N across time levels n ranging from 0 to N. At each discrete space-time node, the non-local Caputo–Fabrizio fractional time derivative of order alpha is expressed as a temporal convolution integral divided into a summation of subintegrals over consecutive time intervals [t_{j-1}, t_j]. Across each subinterval, the continuous velocity time derivative is approximated using a first-order backward difference quotient with local error of order O(Delta t). Integrating the exponential kernel analytically yields an exact algebraic expression involving exponential weighting factors d_{j,n}, transforming historical memory into a computable algebraic sum.

Building on this discrete fractional derivative operator, the research team applies the implicit Crank–Nicholson finite difference scheme to discretize the spatial differential terms of the fractional Burgers equation. To preserve stability and high-order spatial accuracy, the nonlinear advection term u du/dx and the second-order viscous diffusion term nu d^2u/dx^2 are approximated through symmetric averages evaluated at time levels n and n – 1. Specifically, the nonlinear convective term is discretized using an implicit central difference splitting that couples velocity components across both time levels over a 4 Delta x denominator, linearizing the advection product while maintaining second-order spatial consistency. The second-order diffusion term is similarly discretized using standard central difference operators evaluated at both time levels over 2 Delta x^2, while the external source term g(x,t) is approximated by an arithmetic mean. Collecting all unknown velocity terms at time level n on the left-hand side generates a tridiagonal linear algebraic system, ensuring an implicit time-stepping algorithm.

A rigorous theoretical component of the investigation is the mathematical demonstration of unconditional stability for the proposed Crank–Nicholson scheme using discrete Fourier analysis. Defining the round-off error grid quantity rho_i^n as the numerical deviation from the exact solution under homogeneous boundary conditions, the authors derive the corresponding round-off error difference equation. They construct a piecewise grid function across the computational domain and expand it as a complex Fourier series with Fourier coefficients lambda_n(l). According to Parseval’s equality, the discrete l^2 norm of the spatial error vector equals the square sum of its Fourier series coefficients. Substituting the single-mode Fourier harmonic ansatz into the round-off error equation produces an algebraic recurrence connecting successive time levels. Utilizing mathematical induction alongside defined parameters mu_1 and mu_2, the authors prove Proposition 1, establishing that |lambda^n| <= |lambda^0| for all discrete time levels. Applying Parseval’s identity confirms Theorem 2, showing that the scheme is unconditionally stable.

Following the stability demonstration, the authors establish the convergence of their numerical algorithm within the discrete l^2 norm using Taylor series truncation analysis combined with Fourier methods. Substituting the exact analytical solution into the discrete finite difference operator defines the local truncation error R_i^n, which is bounded through Taylor expansions as |R_i^n| <= H_1 (Delta t + Delta x^2), where H_1 denotes a positive constant representing maximum derivative bounds across the grid. Defining the global computational error as epsilon_i^n with zero initial and boundary errors, the resulting error equation is transformed into the Fourier domain by expanding grid error functions and truncation functions into Fourier series. By applying mathematical induction across time levels, the authors establish Proposition 2, showing that Fourier error amplitudes are bounded by H_2 |eta_1|. Connecting this bound to truncation error via Parseval’s equality culminates in Theorem 3, proving that global error satisfies ||epsilon^n||_2 <= H (Delta t + Delta x^2), confirming first-order temporal and second-order spatial convergence.

The authors validate the theoretical convergence rates and computational feasibility of their implicit Crank–Nicholson Caputo–Fabrizio scheme by implementing two distinct benchmark problems in Matlab. The first test problem evaluates the fractional Burgers equation on the unit spatial interval with homogeneous Dirichlet boundary conditions, zero initial condition, kinematic viscosity nu = 1, fractional order alpha = 0.5, and an exact analytical solution u(x,t) = t^2 sin(2 pi x). Testing a spatial mesh size of h = 0.1 alongside varying time step increments at terminal time T = 1 demonstrates close agreement between numerical approximations and exact analytical profiles, with maximum norm absolute errors confirming expected convergence behavior alongside recorded CPU execution times. The second numerical experiment evaluates the governing equation under inhomogeneous boundary conditions u(0,t) = t^2 and u(1,t) = e t^2 with zero initial condition, kinematic viscosity nu = 1, and exact solution u(x,t) = t^2 e^x, demonstrating stable error profiles and minimal computational overhead across fractional orders alpha = 0.3 and alpha = 0.7.

To benchmark the performance of their proposed Crank–Nicholson Caputo–Fabrizio (CNCF) scheme against existing numerical approaches in published literature, the researchers conducted a comparative analysis against a finite difference method coupled with cubic B-spline functions and the Atangana–Baleanu fractional derivative (FDCB). Because both the Caputo–Fabrizio and Atangana–Baleanu formulations utilize non-singular kernels, the authors solved the second benchmark problem under identical conditions: fractional order alpha = 0.2, kinematic viscosity nu = 1, spatial mesh parameter M = 2^4 = 16, and varying temporal step counts N. The comparative error norms reveal that increasing the temporal resolution N leads to systematic error reductions in both numerical frameworks, with the empirical order of convergence in both cases approximating two. The researchers conclude that their implicit finite difference formulation with non-singular kernel derivatives provides an unconditionally stable, accurate, and computationally robust foundation for solving fractional-order Burgers equations, establishing a verified mathematical framework that can subsequently be generalized to multi-dimensional fractional Navier–Stokes fluid flow models.

Subject of Research: Implicit Crank-Nicholson numerical scheme for fractional Burgers equation with Caputo-Fabrizio derivative

Article Title: Implicit numerical technique for solving fractional Navier-Stoks equation with non-singular kernel

Article References: Azami, L., Sheikhani, A. H. R., & Najafi, H. S. (2025). Implicit numerical technique for solving fractional Navier-Stoks equation with non-singular kernel. Advanced Modeling and Simulation in Engineering Sciences, 12(1), Article 23. https://doi.org/10.1186/s40323-025-00300-x

Image Credits: AI Generated

DOI: 10.1186/s40323-025-00300-x

Keywords: Burgers equation, Caputo-Fabrizio fractional derivative, Crank-Nicholson scheme, Navier-Stokes equation, finite difference method, non-singular kernel, unconditional stability, Fourier analysis, traffic flow modeling, kinematic viscosity, advection-diffusion equilibrium, Sobolev space, truncation error, Parseval equality, Atangana-Baleanu derivative, cubic B-spline, Matlab numerical simulation