In this paper, singularly perturbed boundary value problem of fourth order ordinary differential ... more In this paper, singularly perturbed boundary value problem of fourth order ordinary differential equation with a small positive parameter multiplying with the highest derivative of the form εu(4)(x) + p(x)u ′′ (x) + q(x)u(x) = r(x), 0 ≤ x ≤ 1, u(0) = γ0, u(1) = γ1, u ′′ (0) = η0, u ′′ (1) = η1, 0 ≤ ε ≤ 1 is considered. We have developed a numerical technique for the above problem using parametric and polynomial septic spline method. The method is shown to have second and fourth order convergent depending on the choice of parameters involved in the method. Truncation error and boundary equations are obtained. The method is tested on an example and the results are found to be in agreement with the theoretical analysis.
Abstract: In this paper, we have developed parametric septic spline methods, which reduces to ord... more Abstract: In this paper, we have developed parametric septic spline methods, which reduces to ordinary septic spline as the parameter τ → 0 for the numerical solution of fourth order linear and nonlinear two point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. Methods of order two, four and six have been obtained which lead to seven diagonal linear system. Boundary equations for existing orders have been developed and truncation error is obtained. Three numerical illustrations are tabulated to demonstrate the practical usefulness of our methods and comparison is made with known methods.
Hacettepe Journal of Mathematics and Statistics, 2016
In this paper, we report three level implicit method of high accuracy schemes for the numerical s... more In this paper, we report three level implicit method of high accuracy schemes for the numerical solution of fourth order nonhomogeneous parabolic partial dierential equation, that governs the behavior of a vibrating beam. Parametric septic spline is used in space and nite dierence discretization in time. The linear stability of the presented method is investigated. The computed results for three examples are compared wherever possible with those already available in literature which shows the superiority of the proposed method.
Discrete cubic spline technique for solving one-dimensional Bratu’s problem
Asian-European Journal of Mathematics
In this paper, discrete cubic spline method based on central differences is developed to solve on... more In this paper, discrete cubic spline method based on central differences is developed to solve one-dimensional (1D) Bratu’s and Bratu’s type highly nonlinear boundary value problems (BVPs). Convergence analysis is briefly discussed. Four examples are given to justify the presented method and comparisons are made to confirm the advantage of the proposed technique.
In this paper, we propose a spline approach for the numerical solution of fourth order parabolic ... more In this paper, we propose a spline approach for the numerical solution of fourth order parabolic partial differential equation that governs the behavior of a vibrating beam. We have used nonpolynomial cubic tension spline in space and finite difference discretization in time. Class of methods and Stability analysis have been carried out. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.
In this paper, a new three-level implicit method is developed to solve linear and non-linear thir... more In this paper, a new three-level implicit method is developed to solve linear and non-linear third order dispersive partial differential equations. The presented method is obtained by using exponential quartic spline to approximate the spatial derivative of third order and finite difference discretization to approximate the first order spatial and temporal derivative. The developed method is tested on four examples and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, the truncation error and stability analysis of the presented method are investigated, and graphical comparison between analytical and approximate solution is also shown for each example.
In this paper we develop a non-polynomial quintic spline function to approximate the solution of ... more In this paper we develop a non-polynomial quintic spline function to approximate the solution of third order linear and non-linear boundary value problems associated with odd-order obstacle problems. Such problems arise in physical oceanography and can be studied in the framework of variational inequality theory. The class of methods are second and fourth order convergent. End equations of the splines are derived and truncation error is obtained. Two numerical examples are given to illustrate the applicability and efficiency of proposed method. It is shown that the new method gives approximations, which are better than those produced by other methods.
Parametric quintic spline solution of third-order boundary value problems
International Journal of Computer Mathematics, 2012
In this paper, we develop parametric quintic spline function to approximate the solution of third... more In this paper, we develop parametric quintic spline function to approximate the solution of third-order boundary value problems of the form u″′=f(x, u), a≤x≤b, subject to the boundary conditions u(a)=k1, u′(a)=k2 and u(b)=k3. The class of methods are second-, fourth- and sixth-order accurate. End equations of the splines are derived and truncation error is given. Three numerical examples are presented
In this paper, parametric quintic spline method is presented to solve a linear special case sixth... more In this paper, parametric quintic spline method is presented to solve a linear special case sixth order two point boundary value problems with two different cases of boundary conditions. The method presented in this paper has been shown to be second and fourth order accurate. Boundary equations are derived for both the cases of boundary conditions. Convergence analysis of these methods are discussed. The presented method is tested on four numerical examples of linear sixth order boundary value problems. Comparison is made to show the practical usefulness of the presented method.
Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a... more Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a class of numerical methods for the solution of linear sixth-order two-point boundary value problems. Convergence analysis of the methods is discussed through standard procedure. It is shown that the present methods give approximations, which are better than those produced by other splines and domain decomposition methods. Three numerical examples are given to illustrate the practical usefulness of the new approach.
Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a... more Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a class of numerical methods for the solution of linear sixth-order two-point boundary value problems. Convergence analysis of the methods is discussed through standard procedure. It is shown that the present methods give approximations, which are better than those produced by other splines and domain decomposition methods. Three numerical examples are given to illustrate the practical usefulness of the new approach.
In this paper, singularly perturbed boundary value problem of fourth order ordinary differential ... more In this paper, singularly perturbed boundary value problem of fourth order ordinary differential equation with a small positive parameter multiplying with the highest derivative of the form εu(4)(x) + p(x)u ′′ (x) + q(x)u(x) = r(x), 0 ≤ x ≤ 1, u(0) = γ0, u(1) = γ1, u ′′ (0) = η0, u ′′ (1) = η1, 0 ≤ ε ≤ 1 is considered. We have developed a numerical technique for the above problem using parametric and polynomial septic spline method. The method is shown to have second and fourth order convergent depending on the choice of parameters involved in the method. Truncation error and boundary equations are obtained. The method is tested on an example and the results are found to be in agreement with the theoretical analysis.
Abstract: In this paper, we have developed parametric septic spline methods, which reduces to ord... more Abstract: In this paper, we have developed parametric septic spline methods, which reduces to ordinary septic spline as the parameter τ → 0 for the numerical solution of fourth order linear and nonlinear two point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. Methods of order two, four and six have been obtained which lead to seven diagonal linear system. Boundary equations for existing orders have been developed and truncation error is obtained. Three numerical illustrations are tabulated to demonstrate the practical usefulness of our methods and comparison is made with known methods.
Hacettepe Journal of Mathematics and Statistics, 2016
In this paper, we report three level implicit method of high accuracy schemes for the numerical s... more In this paper, we report three level implicit method of high accuracy schemes for the numerical solution of fourth order nonhomogeneous parabolic partial dierential equation, that governs the behavior of a vibrating beam. Parametric septic spline is used in space and nite dierence discretization in time. The linear stability of the presented method is investigated. The computed results for three examples are compared wherever possible with those already available in literature which shows the superiority of the proposed method.
Discrete cubic spline technique for solving one-dimensional Bratu’s problem
Asian-European Journal of Mathematics
In this paper, discrete cubic spline method based on central differences is developed to solve on... more In this paper, discrete cubic spline method based on central differences is developed to solve one-dimensional (1D) Bratu’s and Bratu’s type highly nonlinear boundary value problems (BVPs). Convergence analysis is briefly discussed. Four examples are given to justify the presented method and comparisons are made to confirm the advantage of the proposed technique.
In this paper, we propose a spline approach for the numerical solution of fourth order parabolic ... more In this paper, we propose a spline approach for the numerical solution of fourth order parabolic partial differential equation that governs the behavior of a vibrating beam. We have used nonpolynomial cubic tension spline in space and finite difference discretization in time. Class of methods and Stability analysis have been carried out. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.
In this paper, a new three-level implicit method is developed to solve linear and non-linear thir... more In this paper, a new three-level implicit method is developed to solve linear and non-linear third order dispersive partial differential equations. The presented method is obtained by using exponential quartic spline to approximate the spatial derivative of third order and finite difference discretization to approximate the first order spatial and temporal derivative. The developed method is tested on four examples and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, the truncation error and stability analysis of the presented method are investigated, and graphical comparison between analytical and approximate solution is also shown for each example.
In this paper we develop a non-polynomial quintic spline function to approximate the solution of ... more In this paper we develop a non-polynomial quintic spline function to approximate the solution of third order linear and non-linear boundary value problems associated with odd-order obstacle problems. Such problems arise in physical oceanography and can be studied in the framework of variational inequality theory. The class of methods are second and fourth order convergent. End equations of the splines are derived and truncation error is obtained. Two numerical examples are given to illustrate the applicability and efficiency of proposed method. It is shown that the new method gives approximations, which are better than those produced by other methods.
Parametric quintic spline solution of third-order boundary value problems
International Journal of Computer Mathematics, 2012
In this paper, we develop parametric quintic spline function to approximate the solution of third... more In this paper, we develop parametric quintic spline function to approximate the solution of third-order boundary value problems of the form u″′=f(x, u), a≤x≤b, subject to the boundary conditions u(a)=k1, u′(a)=k2 and u(b)=k3. The class of methods are second-, fourth- and sixth-order accurate. End equations of the splines are derived and truncation error is given. Three numerical examples are presented
In this paper, parametric quintic spline method is presented to solve a linear special case sixth... more In this paper, parametric quintic spline method is presented to solve a linear special case sixth order two point boundary value problems with two different cases of boundary conditions. The method presented in this paper has been shown to be second and fourth order accurate. Boundary equations are derived for both the cases of boundary conditions. Convergence analysis of these methods are discussed. The presented method is tested on four numerical examples of linear sixth order boundary value problems. Comparison is made to show the practical usefulness of the presented method.
Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a... more Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a class of numerical methods for the solution of linear sixth-order two-point boundary value problems. Convergence analysis of the methods is discussed through standard procedure. It is shown that the present methods give approximations, which are better than those produced by other splines and domain decomposition methods. Three numerical examples are given to illustrate the practical usefulness of the new approach.
Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a... more Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a class of numerical methods for the solution of linear sixth-order two-point boundary value problems. Convergence analysis of the methods is discussed through standard procedure. It is shown that the present methods give approximations, which are better than those produced by other splines and domain decomposition methods. Three numerical examples are given to illustrate the practical usefulness of the new approach.
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