A tenth-order non-polynomial spline method for the solutions of two-point boundary value problem ... more A tenth-order non-polynomial spline method for the solutions of two-point boundary value problem u (4) (x) + f (x, u(x)) = 0, u(a) = λ1, u (a) = λ2, u(b) = λ3, u (b) = λ4, is constructed. Numerical method of tenth-order with end conditions of the order 10 is derived. The convergence analysis of the method has been discussed. Numerical examples are presented to illustrate the applications of method, and to compare the computed results with other known methods.
Journal of Linear and Topological Algebra, Sep 1, 2013
A Class of new methods based on a septic non-polynomial spline function for the numerical solutio... more A Class of new methods based on a septic non-polynomial spline function for the numerical solution one-dimensional Bratu's problem are presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Associated boundary formulas are developed. Convergence analysis of these methods is discussed. Numerical results are given to illustrate the efficiency of methods.
In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to so... more In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to solve the nonlinear Volterra integral equation. When the iterations of the Quasilinear technique employed in nonlinear integral equation we obtain a linear integral equation then by using the Non-polynomial spline functions and collocation method the solution of the integral equation can be approximated. Analysis of convergence is investigated. At the end, some numerical examples are presented to show the effectiveness of the method.
International Journal of Mathematical Modelling & Computations, Mar 21, 2015
A Class of new methods based on a septic non-polynomial spline function for the numerical solutio... more A Class of new methods based on a septic non-polynomial spline function for the numerical solution of problems in calculus of variations is presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th. are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Convergence analysis of these methods is discussed. Numerical results are given to illustrate the efficiency of methods and compared with the methods in [23,32-34].
International Journal of Computer Mathematics, Apr 24, 2019
In our knowledge so far, the Non-polynomial Spline functions (NPS) have not been yet applied for ... more In our knowledge so far, the Non-polynomial Spline functions (NPS) have not been yet applied for approximating the integral equations. In this article, we want to use such functions for obtaining numerical solutions of Fredholm integral equations of the second kind. In our approach, the coefficients of the non polynomial spline are obtained by solving a system of linear equations. Finally, these functions are utilized to reduce the Fredholm integral equations to the solution of algebraic equations. Analysis of convergence is investigated. At the end, our test examples are presented to show the effectiveness of the method.
Computational Methods for Differential Equations, 2021
A new six order method developed for the approximation Fredholm integral equation of the second k... more A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we first formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.
A tenth-order non-polynomial spline method for the solutions of two-point boundary value problem ... more A tenth-order non-polynomial spline method for the solutions of two-point boundary value problem u(x) + f(x, u(x)) = 0, u(a) = λ1, u ′′(a) = λ2, u(b) = λ3, u ′′(b) = λ4, is constructed. Numerical method of tenth-order with end conditions of the order 10 is derived. The convergence analysis of the method has been discussed. Numerical examples are presented to illustrate the applications of method, and to compare the computed results with other known methods.
Computational Methods for Differential Equations, 2020
A new six order method developed for the approximation Fredholm integral equation of the second k... more A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we first, formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quintic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.
A tenth-order non-polynomial spline method for the solutions of two-point boundary value problem ... more A tenth-order non-polynomial spline method for the solutions of two-point boundary value problem u (4) (x) + f (x, u(x)) = 0, u(a) = λ1, u (a) = λ2, u(b) = λ3, u (b) = λ4, is constructed. Numerical method of tenth-order with end conditions of the order 10 is derived. The convergence analysis of the method has been discussed. Numerical examples are presented to illustrate the applications of method, and to compare the computed results with other known methods.
Journal of Linear and Topological Algebra, Sep 1, 2013
A Class of new methods based on a septic non-polynomial spline function for the numerical solutio... more A Class of new methods based on a septic non-polynomial spline function for the numerical solution one-dimensional Bratu's problem are presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Associated boundary formulas are developed. Convergence analysis of these methods is discussed. Numerical results are given to illustrate the efficiency of methods.
In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to so... more In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to solve the nonlinear Volterra integral equation. When the iterations of the Quasilinear technique employed in nonlinear integral equation we obtain a linear integral equation then by using the Non-polynomial spline functions and collocation method the solution of the integral equation can be approximated. Analysis of convergence is investigated. At the end, some numerical examples are presented to show the effectiveness of the method.
International Journal of Mathematical Modelling & Computations, Mar 21, 2015
A Class of new methods based on a septic non-polynomial spline function for the numerical solutio... more A Class of new methods based on a septic non-polynomial spline function for the numerical solution of problems in calculus of variations is presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th. are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Convergence analysis of these methods is discussed. Numerical results are given to illustrate the efficiency of methods and compared with the methods in [23,32-34].
International Journal of Computer Mathematics, Apr 24, 2019
In our knowledge so far, the Non-polynomial Spline functions (NPS) have not been yet applied for ... more In our knowledge so far, the Non-polynomial Spline functions (NPS) have not been yet applied for approximating the integral equations. In this article, we want to use such functions for obtaining numerical solutions of Fredholm integral equations of the second kind. In our approach, the coefficients of the non polynomial spline are obtained by solving a system of linear equations. Finally, these functions are utilized to reduce the Fredholm integral equations to the solution of algebraic equations. Analysis of convergence is investigated. At the end, our test examples are presented to show the effectiveness of the method.
Computational Methods for Differential Equations, 2021
A new six order method developed for the approximation Fredholm integral equation of the second k... more A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we first formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.
A tenth-order non-polynomial spline method for the solutions of two-point boundary value problem ... more A tenth-order non-polynomial spline method for the solutions of two-point boundary value problem u(x) + f(x, u(x)) = 0, u(a) = λ1, u ′′(a) = λ2, u(b) = λ3, u ′′(b) = λ4, is constructed. Numerical method of tenth-order with end conditions of the order 10 is derived. The convergence analysis of the method has been discussed. Numerical examples are presented to illustrate the applications of method, and to compare the computed results with other known methods.
Computational Methods for Differential Equations, 2020
A new six order method developed for the approximation Fredholm integral equation of the second k... more A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we first, formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quintic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.
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