Solving the Viscous Burger Equation Using the Hopf Cole Transform

Year 2023, Volume: 2023 Issue: 19, 1 - 9, 03.01.2024

Münevver Tuz

Abstract

In this study, the non-linear Burger equation is discussed, the equation is linearized by using the Hopf-Cole transform. With the application of this transformation, the system is turned into a Cauchy problem, the boundary conditions are created and the solution is made, and the moving wave solutions are obtained. The study of these moving waves has an important place in fluid dynamics, solitary wave found. Its solutions allow us to obtain exact and real solutions for (2+1) dimensional and (3+1) dimensional nonlinear PDE types in Mathematical physics.

Keywords

Nonlinear equation, Hopf-Cole transform, viscosity, analytical solution, linear system.

References

• [1] Bateman H (1915). Some recent researches on the motion of fluids, Monthly Weather Review, 43, 163-170.
• [2] Lighthill M J (1956). Viscocity effects in sound waves of finite amplitude in Batchlor, Survey in Mechanics, Cambridge University Press, Cambridge, 250-351.
• [3] Miller EL (1966). Predictor-corrector studies of Burger’s Model of turbulent flow, M.S. Thesis, University of Delaware, Newark, Delaware.
• [4] Katz JL, Green ML (1986). A Burgers model of interstellar dynamics, Astronomy & Astrophysics, 161, 139-141.
• [5] Öziş T, Aksan EN, Özdeş A (2003). A finite element approach for solution of Burgers equation, Applied Mathematics and Computation, 139, 417-428.
• [6] Benton E, Platzman GW (1972). A table of solutions of the one-dimensional Burgers equations, Quarterly of Applied Mathematics, 30, 195-212.
• [7] Liao W (2008). An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Applied Mathematics and Computation, 206, 755-764. [8] Sari M, Gurarslan G (2009). A sixth-order compact finite difference scheme to the numerical solutions of Burgers’ equation, Applied Mathematics and Computation, 208, 475-483. [9] Dogan A (2004). A Galerkin finite element approach to Burgers’ equation, Applied Mathematics and Computation, 157, 331- 346.
• [10] Ali AHA, Gardner LRT, Gardner GA (1990). A Galerkin approach to the solution of Burgers’ equation, Mathematics Preprint Series, 90.04, University College of North Wales, Bangor.
• [11] Gardner LRT, Gardner GA (1991). B-spline Finite Elements, Mathematics Preprint Series, 91.10.
• [12] Saka B, Dag I (2007). Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation, Chaos Solitons and Fractals, 32, 1125-1137.
• [13] Hopf E (1950). The partial differential equation Ut +UUx = mUxx, Communications on Pure and Applied Mathematics, 3, 201-230.
• [14] Cole JD (1951). On a quasi-linear parabolic equation in aerodynamics, Quarterly of Applied Mathematics, 9, 225-236.
• [15] Dag I, Irk D, Sahin A (2005). B-spline collocation methods for numerical solutions of the Burgers’ equation, Mathematical Problems in Engineering, 5, 521-538.
• [16] Korkmaz A, Dag I (2013). Cubic B-spline differential quadrature methods and stability for Burgers’ equation, Engineering Computations, 30,(3), 320-344.
• [17] Kofman L, Raga AC (1992). Modeling structures of knots in jet flows with the Burgers equation, The Astrophysical Journal, 390, 359-364.
• [18] Zhu G, Wang RH (2009). Numerical solution of Burgers’ equation by cubic B-spline quasi-interpolation, Applied Mathematics and Computation, 208, 260-272.