Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems

Abbas, Muhammad; Majid, Ahmad Abd.; Ismail, Ahmad Izani Md.; Rashid, Abdur

doi:10.1155/2014/849682

Cited by 26 publications

(23 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The approximate solution ( , ) to the exact solution exc ( , ) can be expressed as follows [59][60][61][62][63][64]:…”

Section: Hybrid B-spline Basis Functionmentioning

confidence: 99%

“…Further two equations are included in the resulting system to obtain a unique solution of the problem using the boundary conditions given in (21). The initial vector 0 can be obtained from initial condition given in (2) [59][60][61][62][63][64]. Thus, the resulting system becomes a matrix system of dimension ( +3)×( +3) which is a tridiagonal system that can be solved by Thomas algorithm [65][66][67].…”

Section: Numerical Solution Of the Generalized Burgers-fisher Andmentioning

confidence: 99%

See 1 more Smart Citation

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

Wasim

Abbas

Noor‐ul‐Amin

2018

Mathematical Problems in Engineering

Self Cite

View full text Add to dashboard Cite

In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.

show abstract

“…The approximate solution ( , ) to the exact solution exc ( , ) can be expressed as follows [59][60][61][62][63][64]:…”

Section: Hybrid B-spline Basis Functionmentioning

confidence: 99%

Section: Numerical Solution Of the Generalized Burgers-fisher Andmentioning

confidence: 99%

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

Wasim

Abbas

Noor‐ul‐Amin

2018

Mathematical Problems in Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…The trigonometric B-spline collocation method has attracted attention in the literature and has been used for the numerical solutions of several linear and nonlinear partial differential equations [20][21][22][23][24][25]. The trigonometric B-splines have many geometric properties like local support, smoothness, and capability of handling local phenomena.…”

Section: 2applicationsmentioning

confidence: 99%

Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach

2017

Self Cite

View full text Add to dashboard Cite

This paper presents a new approach and methodology to solve the second order one dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions using the cubic trigonometric B-spline collocation method. The usual finite difference scheme is used to discretize the time derivative. The cubic trigonometric B-spline basis functions are utilized as an interpolating function in the space dimension, with a weighted scheme. The scheme is shown to be unconditionally stable for a range of  values using the von Neumann (Fourier) method. Several test problems are presented to confirm the accuracy of the new scheme and to show the performance of trigonometric basis functions. The proposed scheme is also computationally economical and can be used to solve complex problems. The numerical results are found to be in good agreement with known exact solutions and also with earlier studies.Keywords: Second order one dimensional telegraph equation, Cubic trigonometric B-spline basis functions, Cubic trigonometric B-spline collocation method, Stability * Corresponding Email address: m.abbas@uos.edu.pk 2 1.3.Literature review Several numerical methods have been developed to solve the telegraph equation subject to Dirichlet boundary conditions and the references are in [2,[5][6][7][8]. In [9], two semi-discretization methods based on quartic splines function have been developed to solve the telegraph equations. A class of unconditionally stable finite difference schemes constructed with the help of quartic splines functions has been developed by Liu and Liu [10] for the solution of the telegraph equation. Further several numerical methods have been developed by Dehghan [11][12] in collaboration with different authors. These include the thin plate splines radial basis functions (RBF) for the numerical solution of the telegraph equation [11] and high-order compact finite difference method to solve the telegraph equation [12]. Further details on other numerical methods including interpolating scaling functions [13], radial basis functions [14], quartic B-spline collocation method (QuBSM) [15], cubic B-spline collocation method (CuBSM) [16][17] for the solution of the telegraph equation subject to Dirichlet boundary conditions are in the literature. Thus many numerical methods have been developed to solve the telegraph equation (1) with Dirichlet boundary conditions. Some numerical methods have been developed for numerical solution of the telegraph equation with Neumann boundary conditions. These include methods by Dehghan and Ghesmati [1] who constructed a dual reciprocity boundary integral equation (DRBIE) method in which cubic-radial basis function (C-RBF), thin plate-spline-radial basis function (TPS-RBF) and linear radial basis functions (L-RBF) are utilized for the numerical solution of the telegraph equation with Neumann boundary conditions. Liu and Liu [18] have developed a compact difference unconditionally stable scheme (CDS) to solve the telegraph equation with Neumann boundary conditions. Further, Mi...

show abstract

“…Solutions of the ordinary differential equation, having singular boundary value was presented by a numerical method employed the TCB in the study [12]. Very recently a collocation finite difference scheme based on TCB is developed for integration of one-dimensional hyperbolic equation (wave equation) with non-local conservation condition and the nonclassical diffusion problem with nonlocal boundary condition into a system of linear algebraic equations respectively in the studies [13,14]. A new two-time level implicit technique is proposed for the approximate solution of in the study [14].…”

Section: Introductionmentioning

confidence: 99%

“…Very recently a collocation finite difference scheme based on TCB is developed for integration of one-dimensional hyperbolic equation (wave equation) with non-local conservation condition and the nonclassical diffusion problem with nonlocal boundary condition into a system of linear algebraic equations respectively in the studies [13,14]. A new two-time level implicit technique is proposed for the approximate solution of in the study [14]. Some researches have established types of the B-spline finite element approaches for solving the FE but not with the TCB as far as we search in the literature.…”

Section: Introductionmentioning

confidence: 99%

The Numerical Approach to the Fisher's Equation via Trigonometric Cubic B-spline Collocation Method

Hepson¹,

Dağ²

2017

Communications in Numerical Analysis

View full text Add to dashboard Cite

In this study, we set up a numerical technique to get approximate solutions of Fisher's equation which is integrated fully with help of combination of the trigonometric cubic B-spline functions in space and Crank-Nicolson in time. Numerical results have been presented to show utility of the current algorithm by studying three test problems. Solutions of the fisher equation are shown to model the wave front numerically and interaction of diffusion and reaction is observed for the Fisher eqaution.

show abstract

Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems

Cited by 26 publications

References 42 publications

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach

The Numerical Approach to the Fisher's Equation via Trigonometric Cubic B-spline Collocation Method

Contact Info

Product

Resources

About