Time Fractional Burgers Research Articles

Bivariate Jacobi polynomials depending on four parameters and their effect on solutions of time-fractional Burgers’ equations

The utilization of time-fractional Burgers’ equations is widespread, employed in modeling various phenomena such as heat conduction, acoustic wave propagation, gas turbulence, and the propagation of chaos in non-linear Markov processes. This study introduces a novel pseudo-operational collocation method, leveraging two-variable Jacobi polynomials. These polynomials are obtained through the Kronecker product of their one-variable counterparts, concerning both spatial (x) and temporal (t) domains. The study explores the impact of four parameters (θ,ϑ,σ,ς>−1) on the accuracy of resulting approximate solutions, marking the first examination of such influence. Collocation nodes in a tensor approach are constructed employing the roots of one-variable Jacobi polynomials of varying degrees in x and t. The study delves into analyzing how the distribution of these roots affects the outcomes. Consequently, pseudo-operational matrices are devised to integrate both integer and fractional orders, presenting a novel methodological advancement. By employing these matrices and appropriate approximations, the governing equations transform into an algebraic system, facilitating computational analysis. Furthermore, the existence and uniqueness of the equations under study are investigated and the study estimates error bounds within a Jacobi-weighted space for the obtained approximate solutions. Numerical simulations underscore the simplicity, applicability, and efficiency of the proposed matrix spectral scheme.

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function

Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-integer-order derivatives. In the current research, an effective cubic B-spline collocation method is used to obtain the numerical solution of the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. It is implemented with the help of a θ-weighted scheme to solve the proposed problem. The spatial derivative is interpolated using cubic B-spline functions, whereas the temporal derivative is discretized by the Atangana–Baleanu operator and finite difference scheme. The proposed approach is stable across each temporal direction as well as second-order convergent. The study investigates the convergence order, error norms, and graphical visualization of the solution for various values of the non-integer parameter. The efficacy of the technique is assessed by implementing it on three test examples and we find that it is more efficient than some existing methods in the literature. To our knowledge, no prior application of this approach has been made for the numerical solution of the given problem, making it a first in this regard.

Solving Inverse System of Time-Fractional Burgers Equation

Solving Inverse System of Time-Fractional Burgers Equation

$$L_p$$-solvability and Hölder regularity for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise

$$L_p$$-solvability and Hölder regularity for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise

Numerical approach for time-fractional Burgers’ equation via a combination of Adams–Moulton and linearized technique

Numerical approach for time-fractional Burgers’ equation via a combination of Adams–Moulton and linearized technique

An Efficient Cubic B-Spline Technique for Solving the Time Fractional Coupled Viscous Burgers Equation

The second order Burger’s equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative is addressed using Caputo’s formula, while the CBS technique with the help of a θ-weighted scheme is utilized to discretize the first- and second-order spatial derivatives. The quasi-linearization technique is used to linearize the non-linear terms. The suggested scheme demonstrates unconditionally stable. Some numerical tests are utilized to evaluate the accuracy and feasibility of the current technique.

The Upwind Finite Volume Element Method for Two-Dimensional Time Fractional Coupled Burgers’ Equation

The finite volume element method for approximating a two-dimensional time fractional coupled Burgers' equation is presented. The linear finite volume element method is used for spatial discretization and the upwind technique is used for the nonlinear convective term to get the semi-discrete scheme. Further, the time-fractional derivative term is approximated by using L1 formula and the nonlinear convection term is treated by linearized upwind technique to get the fully discrete scheme. We prove that the semi-discrete scheme is convergent with one-order accuracy in space and the fully discrete scheme is convergent with one-order accuracy both in time and space in L^2-norm. Numerical experiments are presented finally to validate the theoretical analysis.

A Spectral Galerkin Method Based on the Modified Bases for Solving Time-fractional Burgers Equation

A Spectral Galerkin Method Based on the Modified Bases for Solving Time-fractional Burgers Equation

Two linear finite difference schemes based on exponential basis for two-dimensional time fractional Burgers equation

In present paper, we propose tailored finite point method (TFPM) based on exponential basis for solving two-dimensional time fractional Burgers equation. We use the L1 and L1-2 discretization formulas for the Caputo fractional derivative in time direction and use the TFPM to approximate the spatial derivatives. This method with the help of exponential functions fit the properties of the local solution in time and space simultaneously which act as basis functions in the frame of TFPM. We focus on constructing two linear implicit finite difference schemes, i.e. a five-point node centered scheme and a four-point cell-centered scheme based on exponential functions for deriving the numerical solution of given problem. The stability and convergence of fully discrete schemes based on five-point node centered are discussed. At last, we use some numerical examples to show the accuracy and efficiency of the presented schemes.

A high-accuracy computational technique based on $$L2-1_{\sigma }$$ and B-spline schemes for solving the nonlinear time-fractional Burgers’ equation

A high-accuracy computational technique based on $$L2-1_{\sigma }$$ and B-spline schemes for solving the nonlinear time-fractional Burgers’ equation

A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers’ equation

BackgroundThis study proposes an efficient and stable technique based on new hybrid B-spline (HB-spline) functions for the numerical treatment of the Caputo time fractional nonlinear Burgers’ (TFNB) equation. The time derivative is discretized using the definition of the Caputo derivative, whereas HB-spline functions are used to discretize the spatial derivatives. The Rubin–Graves technique is used to linearize the nonlinear terms.ResultsThe performance and efficacy of the established method are tested using three examples. The graphical results represent the smoothness between numerical and exact solutions. The absolute errors are very low as 1{0}^{-4} and 1{0}^{-5}. The convergence rate shows that the proposed method is second-order accurate in space.ConclusionsThe proposed method provides better results than the methods available in the literature. The method yields highly accurate results and can handle large-scale problems, which is the novelty of the present work.

A Second-Order Scheme for the Generalized Time-Fractional Burgers' Equation

Abstract A second-order numerical scheme is proposed to solve the generalized time-fractional Burgers' equation. The time-fractional derivative is considered in the Caputo sense. First, the quasi-linearization process is used to linearize the time-fractional Burgers' equation, which gives a sequence of linear partial differential equations (PDEs). The Crank–Nicolson scheme is used to discretize the sequence of PDEs in the temporal direction, followed by the central difference formulae for both the first and second-order spatial derivatives. The established error bounds (in the L2− norm) obtained through the meticulous theoretical analysis show that the method is second-order convergent in space and time. The technique is also shown to be conditionally stable. Some numerical experiments are presented to confirm the theoretical results.

A Potent Collocation Approach Based on Shifted Gegenbauer Polynomials for Nonlinear Time Fractional Burgers’ Equations

This paper presents a numerical strategy for solving the nonlinear time fractional Burgers's equation (TFBE) to obtain approximate solutions of TFBE. During this procedure, the collocation approach is used. The proposed numerical approximations are supposed to be a double sum of the products of two sets of basis functions. The two sets of polynomials are presented here: a modified set of shifted Gegenbauer polynomials and a shifted Gegenbauer polynomial set. Some specific integers and fractional derivatives are explicitly given as a combination of basis functions to apply the proposed collocation procedure. This method transforms the governing boundary-value problem into a set of nonlinear algebraic equations. Newton's approach can be used to solve the resulting nonlinear system. An analysis of the precision of the proposed method is provided. Various examples are presented and compared to some existing methods in the literature to prove the reliability of the suggested approach.

An approximate solution of a time fractional Burgers’ equation involving the Caputo-Katugampola fractional derivative

The reduced version of the fractional Laplace transform, called the v-Laplac transform, is used in combination with the Adomian decomposition method to generate approximate solutions of the fractional Berger's equation with the Caputo-Katugampola fractional derivative. The effect of the order of the Caputo-Katugampola fractional derivative in Berger's equation is analyzed. The obtained approximate solutions are displayed graphically. The graphs and numerical solutions have demonstrated a tight correspondence between the exact and v-Laplace DM solutions. It is observed that the solutions for various orders u and v display the same behavior and tend to an integer-order problem's solution, confirming the validity of the provided method.

Analytical solutions and mathematical simulation of traveling wave solutions to fractional order nonlinear equations

We investigate the tanh–coth method’s usefulness in understanding space–time fractional nonlinear evolution problems. To use this method and then examine their plots, we take into consideration three significant equations: the space–time fractional Burgers equation, the space–time fractional regularized long wave equation, and the space–time fractional Boussinesq equation. The modified Riemann–Liouville derivatives are used to define the fractional derivative. The approach provides a linearization- or small perturbation-free analytical solution in the form of a convergent series with easily calculable components. For the aforementioned nonlinear fractional equations, we have discovered their precise solutions. We utilize a generalized fractional complex transform to translate these to ordinary differential equations which subsequently resulted into number of exact solutions. The method provides us abundant analytical traveling wave solutions containing fewer number parameters with the aid of Mathematica and MATLAB.

Optimal control of time-fractional stochastic Burgers’ equation driven by mixed fractional Brownian motion

This paper demonstrates the existence, uniqueness, and optimal control of the time-fractional stochastic Burgers’ equation with a deviated argument governed by mixed fractional Brownian motion. The nonlinear deterministic controlled time-fractional Burgers’ equation is subjected to stochastic perturbations to model the memory effects and disturbances that arise in the turbulent flow of liquids. The solvability of the presented system is investigated using stochastic analysis, fractional calculus, and the Nussbaum fixed point theorem. Balder’s theorem is also used to prove the existence of optimal control. At last, an example is presented to demonstrate the developed theory.

A numerical procedure for approximating time fractional nonlinear Burgers–Fisher models and its error analysis

Burger and Fisher diffusion transfer properties and reactions from the characteristics are studied using a non-linear equation. The nonlinear fractional Burgers–Fisher equation (NFB-FE) appears in realistic physical situations such as ultra-slow kinetics, Brownian motion of particles, anomalous diffusion, polymerases of ribonucleic acid and deoxyribonucleic acid, continuous random movement, and formation of wave patterns. The present study focuses on the collocation scheme based on the shifted Chebyshev basis (SCB) and the compact finite difference method to obtain the numerical scheme of the NFB-FE. The simulation model is created in the two steps: Initially, a semi-discrete is formed in a temporal sense, applying a linear approximation with an accuracy order of two. Next, we examine the unconditional stability and the convergence order. In the second stage, the collocation approach based on the SCB of the fourth type is used to discretize the spatial derivative parts and generate the full-discrete scheme.

An implicit nonlinear difference scheme for two-dimensional time-fractional Burgers’ equation with time delay

An implicit nonlinear difference scheme for two-dimensional time-fractional Burgers’ equation with time delay

Numerical Simulation for Generalized Time-Fractional Burgers' Equation With Three Distinct Linearization Schemes

AbstractIn the present study, we examined the effectiveness of three linearization approaches for solving the time-fractional generalized Burgers' equation using a modified version of the fractional derivative by adopting the Atangana-Baleanu Caputo derivative. A stability analysis of the linearized time-fractional Burgers' difference equation was also presented. All linearization strategies used to solve the proposed nonlinear problem are unconditionally stable. To support the theory, two numerical examples are considered. Furthermore, numerical results demonstrate the comparison of linearization strategies and manifest the effectiveness of the proposed numerical scheme in three distinct ways.