Determination of the exact solutions to the inhomogeneous burgers equation with the use of the darboux transformation

Kudryavtsev, A. G.; Sapozhnikov, Oleg A.

doi:10.1134/s1063771011030080

Cited by 11 publications

(4 citation statements)

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“…Maximum absolute errors of (34) subject to (35) and (36) are introduced in Table 1 using J-GL-C method with various choices of , and , and − = = 10; this table indicates that the obtained results are very accurate for a small choice of , while the absolute errors of problem (34) are presented in Table 2 In Figure 1, we see that the approximate solution and the exact solution for = 0.5 of problem (34) coincide with values of parameters listed in their caption. Moreover, the absolute error of problem (34) where = = −1/2 and = 2 is displayed in Figure 2. In Figure 3, we plotted the approximate solutioñ( , ).…”

Section: Examplementioning

confidence: 98%

“…The idea of commutative hypercomplex mathematics and the homotopy perturbation method were combined to investigate solutions of time-delayed Burgers equation by Rostamy and Karimi [33]. The Darboux transformation was described in [34] to determine the exact solutions to the Burgers equation. Caglar and Ucar [35] proposed the nonpolynomial cubic spline scheme to develop a numerical solution of Burgers equation.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Bhrawy

Assas

Alghamdi

2013

Abstract and Applied Analysis

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Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear timedelayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply (− 1) nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters (, > −1), and the resulting equations together with the two-point boundary conditions constitute a system of (− 1) ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of (− 1) ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms.

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Section: Examplementioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Bhrawy

Assas

Alghamdi

2013

Abstract and Applied Analysis

View full text Add to dashboard Cite

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“…The solutions were expanded to curvilinear coordinate systems by Nerney et al [8]. Using Hopf-Cole and Darboux transformations, Kudryavtsev and Sapozhnikov [9] proposed a method to find the exact solution of the inhomogeneous Burgers' equation. Significant efforts have been made over the past few decades to create reliable numerical techniques for handling the Burgers' equation.…”

Section: Introductionmentioning

confidence: 99%

A Comparative Study of the Explicit Finite Difference Method and Physics-Informed Neural Networks for Solving the Burgers’ Equation

Savović,

Ivanović,

Min

2023

Axioms

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The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical solutions, and, while both approaches yield very good agreement, the EFDM results are more closely aligned with the analytical solutions. Since there is good agreement between all of the numerical findings from the EFDM, PINN, and analytical solutions, both approaches are competitive and deserving of recommendation. The conclusions that are provided are significant for simulating a variety of nonlinear physical phenomena, such as those that occur in flood waves in rivers, chromatography, gas dynamics, and traffic flow. Additionally, the concepts of the solution techniques used in this study may be applied to the development of numerical models for this class of nonlinear partial differential equations by present and future model developers of a wide range of diverse nonlinear physical processes.

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“…There are many methods to solve the viscous and inviscid BE such as the Wood [30] and also found the solution of viscous BE. Kudryavtsev and Sapozhnikov [22] used the Darboux transformation to derived the viscous BE. Aswin et al [3] calculated the solution of the viscous BE by using differential quadrature method (DQM).…”

Section: Introductionmentioning

confidence: 99%

Analytical Method for Solving Inviscid Burger Equation

Amir

Awais

Ashraf

et al. 2023

Punjab Univ. j. math.

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In this paper, we use the natural decomposition method (NDM) for solving inviscid Burger equation (BE). The NDM is associated with the Adomain decomposition method (ADM) and the natural transform method. Applying the analytic method, we solved successfully both lin-ear and non-linear partial differential equations. By applying the NDM, we compute the best approximation solution of linear and non-linear par-tial differential equations. In our experiments, we report comparisons with the exact solution.

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Determination of the exact solutions to the inhomogeneous burgers equation with the use of the darboux transformation

Cited by 11 publications

References 10 publications

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

A Comparative Study of the Explicit Finite Difference Method and Physics-Informed Neural Networks for Solving the Burgers’ Equation

Analytical Method for Solving Inviscid Burger Equation

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