Hypercomplex mathematics and HPM for the time-delayed Burgers equation with convergence analysis

Rostamy, Davood; Karimi, Kobra

doi:10.1007/s11075-011-9448-7

Cited by 4 publications

(3 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this work, there is an emphasis on assessing the convergence analysis for the HPM as a simple scheme rather than the RKDG methods on the aforementioned problem. Hence, we try to obtain the classical solutions by a new idea that is the commutative hypercomplex mathematics [21][22][23]. Therefore, this paper is organized as follows:…”

Section: Communicated By S Georgievmentioning

confidence: 99%

“…For a survey on discontinuous Galerkin method, see [20].In this work, there is an emphasis on assessing the convergence analysis for the HPM as a simple scheme rather than the RKDG methods on the aforementioned problem. Hence, we try to obtain the classical solutions by a new idea that is the commutative hypercomplex mathematics [21][22][23]. Therefore, this paper is organized as follows:In Section 2, we introduce hypercomplex mathematics, and in Section 3, we use hypercomplex mathematics to obtain analytical solutions of the mKdV equation.…”

mentioning

confidence: 99%

See 1 more Smart Citation

The general analytical and numerical solution for the modified KdV equation with convergence analysis

Rostamy

Zabihi

2012

Math Methods in App Sciences

View full text Add to dashboard Cite

Communicated by S. GeorgievWe investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge-Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge-Kutta discontinuous Galerkin method by some numerical illustrations.where a and b are constants. The mKdV equation plays a crucial role in applied mathematics and physics as a nonlinear convection and alike diffusion problem. It has been used to consider acoustic waves in certain anhurmonic lattices, Alfven waves in collision less plasma, Schottky barriers transmission lines, models of traffic congestion, electric circuits, and many different phenomena. Some scientists worked about the mKdV equation and obtained solutions for this equation [7][8][9][10][11]. Wazwaz used three methods for solving the mKdV equation in [12] and obtained three sets of complex and soliton solutions of this equation. Yan obtained a new binary traveling wave periodic solutions of the mKdV equation that are based on some possible Jacobi elliptic functions in [8]. In [13,14], Geyikli and Kaya applied decomposition method for solving this equation, and also, Yan extended the Adomian decomposition method to the mKdV equation and obtained numerical Jacobi elliptic function solution with a assumed initial condition in [9].During the last years, the Runge-Kutta discontinuous Galerkin methods (RKDG) have become a very popular technique for the solution of the nonlinear convection diffusion problem [15][16][17][18][19]. For a survey on discontinuous Galerkin method, see [20].In this work, there is an emphasis on assessing the convergence analysis for the HPM as a simple scheme rather than the RKDG methods on the aforementioned problem. Hence, we try to obtain the classical solutions by a new idea that is the commutative hypercomplex mathematics [21][22][23]. Therefore, this paper is organized as follows:In Section 2, we introduce hypercomplex mathematics, and in Section 3, we use hypercomplex mathematics to obtain analytical solutions of the mKdV equation. In Section 4, we give a brief idea of how to construct the RKDG. In section 5, we recall the idea of In the aforementioned, we have that˛,ˇ, , ı, z 0 are the arbitrary constants, the function SnOE: : : is the Jacobi elliptic function that is defined in [24], k is a 4D algebraic basis element, and c is a scale factor.

show abstract

Section: Communicated By S Georgievmentioning

confidence: 99%

mentioning

confidence: 99%

The general analytical and numerical solution for the modified KdV equation with convergence analysis

Rostamy

Zabihi

2012

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…More recently, a differential quadrature scheme for Burgers equations was proposed in [32]. 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.…”

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

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.

show abstract

A soliton solution of the DD-Equation of the Murnaghan’s rod via the commutative hyper complex analysis

Abourabia

Eldreeny²

2022

Partial Differential Equations in Applied Mathematics

View full text Add to dashboard Cite

Hypercomplex mathematics and HPM for the time-delayed Burgers equation with convergence analysis

Cited by 4 publications

References 10 publications

The general analytical and numerical solution for the modified KdV equation with convergence analysis

The general analytical and numerical solution for the modified KdV equation with convergence analysis

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

A soliton solution of the DD-Equation of the Murnaghan’s rod via the commutative hyper complex analysis

Contact Info

Product

Resources

About