In this paper, we present a numerical algorithm for solving linear integro differential Volterra-Friedholm equations by using spline polynomials of degree ninth with six collocation points. The Fredholm-Volterra equation is converted into a system of
first-order linear differential equations, which is solved by applying polynomials and their derivatives with collocation points. The convergence of the proposed method is demonstrated when it is applied to above problem. To test the effectiveness and accuracy of this method, two test problems were resolved where comparisons could be used with other results taken from recent references to the high resolution provided by spline approximations.
In this paper, we develop spline collocation technique for the numerical solution of
general twelfth-order linear boundary value problems (BVPs). This technique based on
polynomial splines from order sixteenth as well as five collocation points at
every
subinterval of BVPs. The method developed not only approximates the solution of BVP,
but its higher order derivatives as well. We show that the spline collocation method is
existent and unique when it is applied into a test problem. Also, its global truncation error
is estimated. Moreover, the purposed spline method when applied to test problems will be
consistent and convergent from sixteenth order. Three numerical examples are given to
illustrate the applicability and efficiency of the new method. Comparisons of our results
with some other methods show that our method is very effective and successful.
In this paper, we use polynomial splines of eleventh degree with three collocation
points to develop a method for computing approximations to the solution and its
derivatives up to ninth order for general linear and nonlinear ninth-order boundary-v
alue
problems (BVPs). The study shows that the spline method with three collocation points
when is applied to these problems is existent and unique. We prove that the proposed
method if applied to ninth-order BVPs is stable and consistent of order eleven, and it
possesses convergence rate greater than six.
Finally, some numerical experiments are presented for illustrating the theoretical
results and by comparing the results of our method with the other methods, we reveal that
the proposed method is better than others.
In this paper, a spline collocation method is developed for finding numerical solutions of general linear eighth-order boundary-value problems (BVPs) and nonlinear eighth-order initial value problems (IVPs). The presented collocation method affords t
he spline solution by the polynomial of degree eleventh which satisfies the BVPs and IVPs at three collocation points. The study shows that the spline collocation method when is applied such this problems is existent and unique. Moreover, the purposed method if applied to these systems will be consistent and the global truncation error equal eleventh.
Numerical results are given for four examples to illustrate the implementation and efficiency of the method. Comparisons of the results obtained by the present method with results obtained by the other methods reveal that the present method is very effective and convenient.
In this paper, we introduce a numerical method for solving systems of high-index differential algebraic equations. This method is based on approximating the exact solution by spline polynomial of degree eight with five collocation points to find the
numerical solution in each step. The study shows that the method when applied to linear differential-algebraic systems with index equal one is stable and convergent of order 8, while it is stable and convergent of order 9-u for index equal u .
Numerical experiments for four test examples and comparisons with other available results are given to illustrate the applicability and efficiency of the presented method
