Research Article
An Integral-like Numerical Approach for Solving Burgers’ Equation
Somrath Kanoksirirath*
Issue:
Volume 13, Issue 2, April 2024
Pages:
17-28
Received:
24 April 2024
Accepted:
15 May 2024
Published:
12 June 2024
DOI:
10.11648/j.pamj.20241302.11
Downloads:
Views:
Abstract: The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ1, ӏ2, and ӏ∞ error norms down to the order of 10−4. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x2 > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction.
Abstract: The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are to...
Show More
Research Article
Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy
Issue:
Volume 13, Issue 2, April 2024
Pages:
29-35
Received:
21 May 2024
Accepted:
7 June 2024
Published:
19 June 2024
Abstract: Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)3=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.
Abstract: Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. ...
Show More
