Numerical approach for solving diffusion problems using cubic B-spline collocation method

Gupta, Bharti; Kukreja, V. K.

doi:10.1016/j.amc.2012.08.053

Cited by 11 publications

(4 citation statements)

References 27 publications

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“…In recent years, B-splines collocation method has been studied extensively for the solution of various problems involving differential equations. For exemple; used for the resolution of the hyperbolic telegraph equation [28], diffusion problems [4], convection-diffusion equations [29], nonlinear parabolic partial differential equations [30], Burgers equation [36], time fractional gas dynamics equation [14], generalized BlackScholes equation governing option pricing [31] and class of partial integrodifferential equation [17].…”

Section: Collocation Methodmentioning

confidence: 99%

Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method

Agoujil¹,

Bouhamidi²,

Chergui³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - S

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The complexity and non-linearity of flow phenomena are explained by numerous criteria, including the interactions of the large number of vehicles occupying the road, which influence the road density. This density under certain conditions, leads to traffic congestion which has dangerous effects on the environment such as; resources consumption; noise and the effect caused by greenhouse gas emissions of the CO 2 and other pollutants. In this paper we consider working in an uniform, homogeneous road where the traffic is described by the Lighthill Whitham-Richard (LWR) model resolved using a cubic B-spline collocation scheme in space and an implicit Runge Kutta scheme in time. We also shed light on the relation between vehicle occupancy and vehicle emissions.

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Section: Collocation Methodmentioning

confidence: 99%

Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method

Agoujil¹,

Bouhamidi²,

Chergui³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - S

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“…Research on the B-Spline Cubic Collocation method has been discussed by many researchers. In 2012, Gupta and Kukreja studied the numerical solution of diffusion problems by comparing the B-Spline Cubic Collocation method and finite difference methods [7]. One year later, Bhatia and Mittal (2013) studied the numerical scheme for solving the onedimensional second order of the hyperbolic telegraph equation using the B-Spline Cubic Collocation method for spatial variables and the four-fifth order Runge-Kutta method for time variables with Dirichlet boundary conditions [2].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Burgers Equation using Cubic B-Spline Collocation Method and Neumann Boundary Condition

Lezani,

Habibah

2023

IJMA

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Burgers equation is one of the nonlinear differential equations which is generally difficult to determine its analytical solution, so it is necessary to do a numerical approach. This article discusses the numerical solution of the Burgers equation using the Cubic B-Spline Collocation method. The first step is to derive the numerical scheme using the Cubic B-Spline Collocation method for the space variable and the Crank-Nicholson method for the time variable. Furthermore, based on von Neumann stability analysis, it is obtained that the numerical scheme of Burgers equation is unconditionally stable. By performing numerical simulations using different and step sizes, it can be shown that the absolute value of the resulting error will be smaller for the step sizes of and which is getting smaller.

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“…Mathematicians chose the term spline to refer to the piecewise polynomial functions used to create smooth curves. This method of constructing curves is used in data interpolation [12,20], in computer design software to create smooth surfaces [13,18], and partial differential equations to find numerical solutions [9,14].…”

Section: Introductionmentioning

confidence: 99%

Integer Generalized Splines on Cycles

Handschy,

Melnick,

Reinders

2014

Preprint

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Let G be a graph whose edges are labeled by positive integers. Label each vertex with an integer and suppose if two vertices are joined by an edge, the vertex labels are congruent to each other modulo the edge label. The set of vertex labels satisfying this condition is called a generalized spline. Gilbert, Polster, and Tymoczko recently defined generalized splines based on work on polynomial splines by Billera, Rose, Haas, Goresky-Kottwitz-Machperson, and many others. We focus on generalized splines on n-cycles. We construct a particularly nice basis for the module of splines on n-cycles. As an application, we construct generalized splines on star graphs, wheel graphs, and complete graphs.

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Numerical approach for solving diffusion problems using cubic B-spline collocation method

Cited by 11 publications

References 27 publications

Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method

Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method

Numerical Solution of Burgers Equation using Cubic B-Spline Collocation Method and Neumann Boundary Condition

Integer Generalized Splines on Cycles

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