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Solving the Viscous Burger Equation Using the Hopf Cole Transform

Yıl 2023, Cilt: 2023 Sayı: 19, 1 - 9, 03.01.2024

Münevver Tuz

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • [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.

Yıl 2023, Cilt: 2023 Sayı: 19, 1 - 9, 03.01.2024

Münevver Tuz

Öz

Kaynakça

  • [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.
Toplam 16 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Yazılım Testi, Doğrulama ve Validasyon
Bölüm Research Article
Yazarlar

Münevver Tuz 0000-0002-9620-247X

Erken Görünüm Tarihi 27 Aralık 2023
Yayımlanma Tarihi 3 Ocak 2024
Gönderilme Tarihi 22 Ekim 2022
Kabul Tarihi 12 Haziran 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 2023 Sayı: 19

Kaynak Göster

APA Tuz, M. (2024). Solving the Viscous Burger Equation Using the Hopf Cole Transform. Journal of New Results in Engineering and Natural Sciences, 2023(19), 1-9.