Analytical and numerical solutions for the nonlinear Burgers and advection–diffusion equations by using a semi-analytical iterative method

AL-Jawary, M. A.; Azeez, Mustafa Mahmood; Radhi, Ghassan Hasan

doi:10.1016/j.camwa.2018.04.010

Cited by 28 publications

(16 citation statements)

References 24 publications

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“…The method shows a very accurate and high order of convergence, reliable and effectual and also optional in any restrictive assumptions for non-linear terms. This can be supported in [14][15][16].…”

Section: Introductionmentioning

confidence: 69%

The Semi Analytics Iterative Method for Solving Newell-Whitehead-Segel Equation

Latif¹,

Selamat²,

Rosli³

et al. 2020

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Newell-Whitehead-Segel (NWS) equation is a nonlinear partial differential equation used in modeling various phenomena arising in fluid mechanics. In recent years, various methods have been used to solve the NWS equation such as Adomian Decomposition method (ADM), Homotopy Perturbation method (HPM), New Iterative method (NIM), Laplace Adomian Decomposition method (LADM) and Reduced Differential Transform method (RDTM). In this study, the NWS equation is solved approximately using the Semi Analytical Iterative method (SAIM) to determine the accuracy and effectiveness of this method. Comparisons of the results obtained by SAIM with the exact solution and other existing results obtained by other methods such as ADM, LADM, NIM and RDTM reveal the accuracy and effectiveness of the method. The solution obtained by SAIM is close to the exact solution and the error function is close to zero compared to the other methods mentioned above. The results have been executed using Maple 17. For future use, SAIM is accurate, reliable, and easier in solving the nonlinear problems since this method is simple, straightforward, and derivative free and does not require calculating multiple integrals and demands less computational work.

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Section: Introductionmentioning

confidence: 69%

The Semi Analytics Iterative Method for Solving Newell-Whitehead-Segel Equation

Latif¹,

Selamat²,

Rosli³

et al. 2020

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“…where L = ⌊ 100 Δx ⌋ . The jump probabilities for the boundary points must be computed using the single point quadrature form, Equation (28), so as to avoid the need for an additional point exterior to the boundary. Thus,…”

Section: Example 1: Dirichlet Boundary Conditionsmentioning

confidence: 99%

“…Once again, the jump probabilities for the ghost points must be computed using the single point quadrature form, Equations (27) and (28), so as to avoid the need for an additional ghost point.…”

Section: Example 2: Neumann Boundary Conditionsmentioning

confidence: 99%

Numeric solution of advection–diffusion equations by a discrete time random walk scheme

Angstmann

Henry

Jacobs

et al. 2019

Numerical Methods Partial

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Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for nonlinear advection–diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.

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“…A semi-analytical iterative technique was proposed by Temimi and Ansari (TAM) (Temimi & Ansari, 2011) to solve nonlinear problems. It has been used to solve many differential equations, such as second-order nonlinear ODEs that occur in physics (Al-Jawary, Adwan & Radhi, 2020), nonlinear Burgers advection-diffusion equations (Al-Jawary, Azeez & Radhi, 2018), Fornberg-Whitham equation (Almjeed, 2018), solving chemistry problems (Al-Jawary & Raham, 2017), Convective Straight and Radial Fins with temperature dependent thermal conductivity problems (Abdul Nabi & Al-Jawary, 2019), nonlinear thin film flow problems (Al-Jawary, 2017) and Fokker-Planck's equations (Al-Jawary, Radhi, & Ravnik, 2017). In addition, an alternative iterative method called Banach Contraction Principle (BCP) by Varsha…”

Section: Introductionmentioning

confidence: 99%

Analytic and numerical solutions for linear and nonlinear multidimensional wave equations

Adwan

AL-Jawary

Tibaut

et al. 2020

Arab Journal of Basic and Applied Sciences

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We develop three reliable iterative methods for solving the nonlinear 1D, 2D and 3D second-order wave equation and compare the results with a discretization-based solver. The iterative Tamimi-Ansari method (TAM), Daftardar-Jafari method (DJM) and the Banach contraction method (BCM) are used to obtain the exact solution for linear equations. For nonlinear equations and practical problems, however, one obtains the approximate solutions that converge to the exact solution, if one exists. The convergence analysis of the three methods is shown using the fixed-point theorem. The methods prove to be quite efficient and well suited to solve this kind of problems. We present several examples that demonstrate the accuracy and efficiency of the methods. We also compare the methods with a method based on discretization (Boundary Domain Integral Method (BDIM)). The BDIM uses a standard domain grid and discretizes the integral form of the governing equations. The iterative methods were developed with MathematicaV R 10, while BDIM is a proprietary development. ARTICLE HISTORY

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Analytical and numerical solutions for the nonlinear Burgers and advection–diffusion equations by using a semi-analytical iterative method

Cited by 28 publications

References 24 publications

The Semi Analytics Iterative Method for Solving Newell-Whitehead-Segel Equation

The Semi Analytics Iterative Method for Solving Newell-Whitehead-Segel Equation

Numeric solution of advection–diffusion equations by a discrete time random walk scheme

Analytic and numerical solutions for linear and nonlinear multidimensional wave equations

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