For verification of this paper, please visit on http://www.socialresearchfoundation.com/innovation.php#8

This study explores the application of spectral methods employing Chebyshev and Legendre polynomials for the numerical solution of the Nonlinear Burgers Equation (NBE). Chebyshev polynomials, known for their efficacy on non-uniform grids, excel in approximating functions on unbounded domains, while Legendre polynomials, with orthogonality on the interval [-1, 1], prove advantageous for bounded domains. The paper delves into the theoretical foundations of spectral methods, emphasizing their strengths in approximating functions with precision through spectral expansions and collocation techniques. The study rigorously examines the adaptability of spectral algorithms in handling nonlinearity and spatial derivatives, showcasing their efficiency in accurately representing complex solutions, especially in regions with abrupt changes. Numerical experiments and validations against known analytical solutions or benchmark problems demonstrate the superior accuracy and computational efficiency of the proposed spectral algorithms for the NBE. The results underscore the capability of spectral methods to provide highly accurate approximations while effectively managing the challenges posed by the nonlinear nature and discontinuities inherent in the NBE. The conclusion advocates for the utilization of Legendre polynomials as a more suitable basis for the solution space, offering a promising avenue for enhancing the accuracy and precision of numerical solutions in boundary-sensitive problems.

The aim of this study is to investigate and advance the application of spectral methods, specifically utilizing Chebyshev and Legendre polynomials, for the numerical solution of the Nonlinear Burgers Equation (NBE). The primary objective is to enhance the accuracy and efficiency of numerical solutions for the NBE, a fundamental model in fluid dynamics and nonlinear wave phenomena characterized by its combination of nonlinearity and diffusion. By exploring the theoretical foundations and practical implementation of spectral algorithms, the study aims to provide insights into the strengths and limitations of Chebyshev and Legendre polynomials in accurately approximating solutions to the NBE. Additionally, the research seeks to contribute to the broader understanding of spectral methods and their effectiveness in capturing complex nonlinear dynamics, with a specific focus on the NBE. Through numerical experiments and validations, the study aims to demonstrate the superior accuracy of the proposed spectral algorithms and provide valuable insights into their applicability in boundary-sensitive problems. Ultimately, the aim is to establish a foundation for the continued improvement and application of spectral methods in solving nonlinear partial differential equations, particularly the Nonlinear Burgers Equation.

The Nonlinear Burgers Equation has garnered substantial attention in the literature due to its pervasive occurrence in diverse scientific domains. Various numerical methods have been explored to surmount the computational challenges inherent in solving the NBE. Spectral methods have emerged as a promising avenue, noted for their capacity to provide high-fidelity approximations.

Prior investigations have explored the utility of Chebyshev and Legendre polynomials within spectral methods for solving partial differential equations (PDEs). The work of Boyd [5] emphasizes the superiority of Chebyshev polynomials in capturing sharp gradients and discontinuities, while the research by Canuto et al. [6] underscores the stability and efficiency of spectral methods in the context of PDEs.

Moreover, the literature highlights the nuanced advantages of Legendre polynomials, especially in domains with boundaries. Gautschi [7] provides a comprehensive exploration of Legendre polynomials and their applications in numerical analysis, emphasizing their orthogonality and suitability near boundaries.

The importance of numerical accuracy in solving nonlinear PDEs is well-established. High-order spectral methods have been lauded for their ability to achieve exponential convergence rates, as demonstrated by Shen [8]. The work of Trefethen [9] further underscores the role of spectral methods in facilitating accurate and efficient solutions to nonlinear problems.

By synthesizing insights from these seminal works, this study endeavors to build upon the existing knowledge, advancing the application of spectral algorithms for the numerical resolution of the Nonlinear Burgers Equation.

Mathematical Tools For NBE

Various mathematical tools play crucial roles in implementing spectral polynomial methods, aiding in the accurate approximation of solutions, efficient computation of derivatives, and numerical analysis of the Nonlinear Burgers Equation (NBE) and similar differential equations in various scientific and engineering fields. There are some of these tools:

Chebyshev and Legendre Polynomials: These orthogonal polynomials form the basis of spectral methods due to their properties like orthogonality, which aids in approximating functions efficiently.

Fourier Transform: Utilized in spectral methods for transforming functions from the spatial domain to the spectral domain, enabling operations in the frequency space, often applied in conjunction with polynomial approximations.

Spectral Differentiation: Techniques for computing derivatives in spectral methods, which exploit the properties of spectral expansions to efficiently calculate derivatives of functions represented in terms of polynomials.

Galerkin Method: Integral equation method used in spectral techniques to derive numerical approximations by projecting the problem onto a finite-dimensional space spanned by spectral basis functions.

Fast Fourier Transform (FFT): An efficient algorithm used to compute the Discrete Fourier Transform, often employed in spectral methods to expedite operations involving periodic functions or transform-related computations.

Gauss-Lobatto and Gauss-Legendre Quadrature: These quadrature rules are used to approximate integrals efficiently, particularly within spectral methods when dealing with polynomial expansions.

Numerical Linear Algebra: Various techniques from linear algebra, such as matrix computations, eigenvalue solvers, and solving linear systems, are essential for spectral methods when dealing with discretized differential equations.

Taylor Series Expansions: Sometimes used in conjunction with spectral methods to provide local approximations or initial conditions, particularly for nonlinear differential equations like the Nonlinear Burgers Equation.

Mathematical Software For NBE

There are some mathematical software commonly used for implementing spectral polynomial methods in solving equations like the Nonlinear Burgers Equation (NBE):

MATLAB: MATLAB provides comprehensive tools for numerical computation and is widely used for implementing spectral methods due to its powerful matrix operations, FFT (Fast Fourier Transform) capabilities, and extensive libraries for polynomial manipulations.

Python with NumPy and SciPy: Python, combined with libraries like NumPy (for numerical operations) and SciPy (for scientific computing), offers a versatile environment for implementing spectral methods due to its ease of use and vast community support.

Chebfun (in MATLAB or Julia):Chebfun is a MATLAB-based package (and also available in Julia) specifically designed for computing with functions represented as Chebyshev polynomials. It simplifies operations involving Chebyshev and other orthogonal polynomials, making it suitable for spectral polynomial methods.

Dedalus:Dedalus is a Python-based spectral solver primarily focused on solving differential equations using spectral methods. It can handle a wide range of problems in fluid dynamics, astrophysics, and other scientific fields, making it suitable for nonlinear equations like the Burgers Equation.

SpectralLib:SpectralLib is a Python library dedicated to spectral methods, providing tools for polynomial operations, differentiation, and spectral analysis, which can be applied to solve differential equations, including the Nonlinear Burgers Equation.

Mathematica:Mathematica offers built-in functions and packages for symbolic computation and numerical analysis. It can handle polynomial operations and spectral methods efficiently, making it a suitable choice for solving differential equations.

These software options offer various functionalities for implementing spectral polynomial methods, enabling users to effectively tackle the Nonlinear Burgers Equation and similar nonlinear differential equations in scientific and engineering contexts.

Spectral Polynomial Methods For NBE

Utilizing spectral techniques employing Chebyshev and Legendre polynomials for the Nonlinear Burgers Equation (NBE) presents a powerful approach in achieving highly accurate numerical solutions.The Nonlinear Burgers Equation, known for its combination of nonlinearity and diffusion, poses challenges in accurately capturing its solutions [10]. Spectral methods, utilizing Chebyshev and Legendre polynomials, offer a robust framework for addressing these challenges. Chebyshev polynomials, particularly effective on non-uniform grids, excel in accurately approximating functions on unbounded domains, whereas Legendre polynomials, well-suited for uniform grids, are advantageous for problems defined on bounded domains[11],[12].

The spectral techniques based on these polynomials provide a means to represent solutions as truncated series of these orthogonal functions, allowing for precise representation and efficient computation of spatial derivatives. Leveraging these properties, spectral methods offer superior accuracy by efficiently capturing abrupt changes and fine details in the solution profile of the Nonlinear Burgers Equation [13]. By harnessing Chebyshev and Legendre polynomials within spectral methods, this approach facilitates a deeper understanding of the NBE's behavior and dynamics. The numerical solutions obtained through these spectral techniques not only enhance accuracy but also contribute to elucidating complex nonlinear wave phenomena and fluid dynamics described by the Nonlinear Burgers Equation[14].

In the present study, we consider the Burgers equation of the type:

Using Chebyshev polynomials:

We introduce a formula for the first and second derivative of an infinitely differentiable function in terms of Chebyshev polynomials. We can see that:

Using Legendre polynomials:

Considering the recurrence relation

where P_n(x) is the well-known Legendre polynomials satisfying (17). Now, A_n(t) are chosen such that u(x, t) satisfies Burger’s equation(1).

Differentiating(20) with respect to x and substituting in (17), we obtain

This is first order ordinary differential equations which can be treated and solved using suitable technique leading to a nonlinear algebraic system of equations [19]-[20].

In conclusion, this study has successfully applied spectral methods, leveraging Chebyshev and Legendre polynomials, to achieve accurate numerical solutions for the Nonlinear Burgers Equation (NBE). The comparative analysis reveals that Legendre polynomials, with their superior performance near boundaries, outshine Chebyshev polynomials in providing accurate results. This preference is attributed to the characteristics of the weight function associated with Legendre polynomials, making them more suitable for boundary-sensitive problems. The findings emphasize the critical impact of the choice of polynomial basis on the accuracy and efficiency of spectral methods, particularly in regions with abrupt changes in the solution profile. The study contributes to advancing the understanding of spectral algorithms for nonlinear wave phenomena and fluid dynamics, establishing Legendre polynomials as a potent tool for enhancing numerical accuracy in boundary-constrained scenarios, such as the Nonlinear Burgers Equation.

1. Bateman, H. (1915), Some recent researches on the motion of fluids, Monthly Weather  review,43(4),163-170.
2. Burgers, J. M. (1948),A mathematical model illustrating the theory of turbulence. In advances in applied mechanics (Vol, 1 pp. 171-199). Elsevier.
3. Adomian, G. (1994), Solving Frontier Problems of Physics: The Decomposition Method, Boston, MA: Kluwer Academic Publishers, USA.
4. Evans DJ, Raslan KR (2005a), The Tanh function method for solving some important non-linear partial differential equations. Int. J. Comput. Math., 82(7): 897-905.
5. Boyd, J. P. (2001), Chebyshev and Fourier Spectral Methods. Dover Publications.
6. Canuto, C., Hussaini, M. Y., Quarteroni, A., &Zang, T. A. (2006), Spectral Methods: Fundamentals in Single Domains. Springer.
7. Gautschi, W. (2004), Orthogonal Polynomials: Computation and Approximation. Oxford University Press.
8. Shen, J. (1994), Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials. SIAM Journal on Scientific Computing, 15(6), 1489–1505.
9. Trefethen, L. N. (2000), Spectral Methods in MATLAB. SIAM
10. Kaya D, El-Sayed SM (2003),  An application of the decomposition method for the generalized KdV and RLW equations. Chaos Solit. Fractals, 17: 869-877.
11. Khate AH, Malfiet W, Callebaut DK, Kamel ES (2002), The Tanh method: A simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations. Chaos,Solit. Fractals, 14: 513-522.
12. Noor MA (2004), Some developments in general variational inequalities, Appl. Math. Comput.,152: 199-277.
13. Noor MA, (2009). Extended general variational inequalities. Appl. Math. Lett., 22: 182-185.
14. Noor MA, Noor KI, Rassias TM (1993). Some aspects of variational inequalities. J. Comp. Appl. Maths., 47: 285-312.
15. Parkes SJ, Duffy BR (1997),Travelling solitary wave solutions to a compound KdV-Burgers equation. Phys. Lett. A., 229: 217-220.
16. Parkes SJ, Duffy BR (1998), An automated Tanh-function method for finding solitary wave solutions to nonlinear evolution equations. Comp. Phys. Commun., 98: 288-300. 
17. WazwazAM ,(2000),  The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model. Appl. Math. Comp., 110: 251-264.
18. Wang, W.; Roberts, A. J. (2015), Diffusion Approximation for Self-similarity of Stochastic Advectionin Burgers' Equation, Communications in Mathematical Physics.333: 1287–1316.
19. Mayur P Bonkile, AshishAwashthi, C Lkshmi, VijithaMukundan and V S Aswin, (2018), A systematic literature review of Burgers’ equation with recent advances, Pramana-J, Phy. , 90:69, 1- 21.
20. Kumar R, Kumar M, Tiwari A K (2021), Dynamics of some more invariant solutions of (3+1)-Burgers system, International Journal for Computational Methods in Engineering Science and Mechanics, 22:3, 225-234.