Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods

Mittal, R. C.; Dahiya, Sumita

doi:10.1016/j.camwa.2015.04.022

Cited by 17 publications

(4 citation statements)

References 34 publications

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“…Burgers' equation where MCB spline was used in space and SSP-RK43 scheme was implemented to solve a reduced system of ODE. Mittal and Dahiya [35] used the concept of MCB-DQM to solve the hyperbolic diffusion equation numerically, and a reduced set of ODE was dealt with by the SSP-RK43 scheme. Jiwari et al [36] implemented the notion of polynomial DQM to get the numerical solution of 2D sine-Gordon solitons, where the RK 4 method was implemented to solve the obtained system of ODE.…”

Section: Coupled Schrödinger Equation In Two Dimensionsmentioning

confidence: 99%

Numerical Approximation of One- and Two-Dimensional Coupled Nonlinear Schrödinger Equation by Implementing Barycentric Lagrange Interpolation Polynomial DQM

Joshi

Kapoor

Bhardwaj

et al. 2021

Mathematical Problems in Engineering

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In this paper, a new numerical method named Barycentric Lagrange interpolation-based differential quadrature method is implemented to get numerical solution of 1D and 2D coupled nonlinear Schrödinger equations. In the present study, spatial discretization is done with the aid of Barycentric Lagrange interpolation basis function. After that, a reduced system of ordinary differential equations is solved using strong stability, preserving the Runge-Kutta 43 method. In order to check the accuracy of the proposed scheme, we have used the formula of L ∞ error norm. The matrix stability analysis method is implemented to test the proposed method’s stability, which confirms that the proposed scheme is unconditionally stable. The present scheme produces better results, and it is easy to implement to obtain numerical solutions of a class of partial differential equations.

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Section: Coupled Schrödinger Equation In Two Dimensionsmentioning

confidence: 99%

Numerical Approximation of One- and Two-Dimensional Coupled Nonlinear Schrödinger Equation by Implementing Barycentric Lagrange Interpolation Polynomial DQM

Joshi

Kapoor

Bhardwaj

et al. 2021

Mathematical Problems in Engineering

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“…To enumerate the coefficients a 1 ð Þ ik , we fix y-axis in equation ( 3) (Mittal and Dahiya, 2015). "Thomas algorithm" is used to solve the resultant tri-diagonal system and the weighting coefficients a 1 ð Þ 11 ; a 1 ð Þ 12 ; :::::; a 1 ð Þ 1N are computed.…”

Section: Determination Of Weighting Coefficientsmentioning

confidence: 99%

A comparative study of modified cubic B-spline differential quadrature methods for a class of nonlinear viscous wave equations

Mittal

Dahiya

2018

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Purpose In this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by differential quadrature methods (DQM) using a modification of cubic B-spline functions. Two numerical schemes are proposed and compared to achieve numerical approximations for the solutions of nonlinear viscous wave equations. Design/methodology/approach Two schemes are adopted to reduce the given system into two systems of nonlinear first-order partial differential equations (PDE). For each scheme, modified cubic B-spline (MCB)-DQM is used for calculating the spatial variables and their derivatives that reduces the system of PDEs into a system of nonlinear ODEs. The solutions of these systems of ODEs are determined by SSP-RK43 scheme. The CPU time is also calculated and compared. Matrix stability analysis has been performed for each scheme and both are found to be unconditionally stable. The results of both schemes have been extensively discussed and compared. The accuracy and reliability of the methods have been successfully tested on several examples. Findings A comparative study has been carried out for two different schemes. Results from both schemes are also compared with analytical solutions and the results available in literature. Experiments show that MCB-DQM with Scheme II yield more accurate and reliable results in solving viscous wave equations. But Scheme I is comparatively less expensive in terms of CPU time. For MCB-DQM, less depository requirements lead to less aggregation of approximation errors which in turn enhances the correctness and readiness of the numerical techniques. Approximate solutions to the two-dimensional nonlinear viscous wave equation have been found without linearizing the equation. Ease of implementation and low computation cost are the strengths of the method. Originality/value For the first time, a comparative study has been carried out for the solution of nonlinear viscous wave equation. Comparisons are done in terms of accuracy and CPU time. It is concluded that Scheme II is more suitable.

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“…Traditional tumour growth models are based on the diffusion equation (Preziosi, 2003;Swanson et al, 2003;Konukoglu et al, 2010;Painter and Hillen, 2013), as in heat diffusion problems, thermal conduction and dispersion of pollutants represented by the Fourier parabolic linear equation, which basically propagates information through space instantly with infinite velocity (Lurie and Belov, 2020). However, this equation has some shortcomings as it does not describe inertial effects and there are experimental evidences that the diffusive process takes place with finite velocity (Mittal and Dahiya, 2015). This behaviour is known as the 'Heat Conduction Paradox' and contradicts the so-called principle of causality, which states that information cannot travel faster than a finite velocity (Schwarzwälder, 2015).…”

Section: Introductionmentioning

confidence: 99%

A second dynamic population invasion study from reactive telegraph equation and boundary element formulation – a numerical assay about tumour growth in vitro

Pettres

Cirilo

2022

J. Biotechnol. Biodivers.

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This paper is a continuation of a study already carried out on the use of the reactive-telegraph equation to analyse problems of population dynamics based on a formulation of the boundary element method (BEM). In this paper, the numerical model simulates the evolution of a tumour as a problem of population density of cancer cells from different reactive terms coupled to the reactive-telegraph equation to describe the growth and distribution of the population, similar to the two-dimensional in vitro tumour growth experiment. The mathematical model developed is called D-BEM, uses a time independent fundamental solution and the finite difference method is combined with BEM to approximate the time derivative terms and the Gaussian quadrature is used to calculate the domain integrals. The solution of the system nonlinear of equations is based on the Gaussian elimination method and the stability of the proposed formulation was verified. As the telegraph equation has a wave behaviour, a phase change phenomenon that can lead to the appearance of negative population density may occur, an algorithm was developed to guarantee the solution's positivity. Important results were obtained and demonstrate the effect of the delay parameter on tumour growth. In one of the tested cases, the results indicated an oscillatory behaviour in the tumour growth when the delay parameter assumed increasing values. The results of numerical simulations that sought to represent tumour growth, as well as the entire formulation of the boundary elements are presented below.

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Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods

Cited by 17 publications

References 34 publications

Numerical Approximation of One- and Two-Dimensional Coupled Nonlinear Schrödinger Equation by Implementing Barycentric Lagrange Interpolation Polynomial DQM

Numerical Approximation of One- and Two-Dimensional Coupled Nonlinear Schrödinger Equation by Implementing Barycentric Lagrange Interpolation Polynomial DQM

A comparative study of modified cubic B-spline differential quadrature methods for a class of nonlinear viscous wave equations

A second dynamic population invasion study from reactive telegraph equation and boundary element formulation – a numerical assay about tumour growth in vitro

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