A quartic C3-spline collocation method for solving second-order initial value problems

Sallam, S.; Karaballi, A. A.

doi:10.1016/s0377-0427(96)00082-9

Cited by 17 publications

(2 citation statements)

References 16 publications

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“…Spline solutions of IVPs/BVPs have been considered as an efficient tool for the last two decades, see [13,2,21,19,18,15]. It is to be noted that the spline solutions of higher order BVPs have been proved/observed to be second order convergent.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical cascade model leading to 7-th order initial value problem

Akram

Beck

2015

Applied Numerical Mathematics

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In turbulent flows, local velocity differences often obey a cascade-like hierarchical dynamics, in the sense that local velocity differences at a given scale k are driven by deterministic and random forces from the next-higher scale k − 1. Here we consider such a hierarchically coupled model with periodic boundary conditions, and show that it leads to an N-th order initial value problem, where N is the number of cascade steps. We deal in detail with the case N = 7 and introduce a non-polynomial spline method that solves the problem for arbitrary driving forces. Several examples of driving forces are considered, and estimates of the numerical precision of our method are given. We show how to optimize the numerical method to obtain a truncation error of order O (h 5 ) rather than O (h 2 ), where h is the discretization step.

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Section: Introductionmentioning

confidence: 99%

Hierarchical cascade model leading to 7-th order initial value problem

Akram

Beck

2015

Applied Numerical Mathematics

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“…This is important when the solution of the boundary value problem is required at different locations in the interval [a, b ]. This approach has added advantage that it not only provides continuous approximations to y(x), but also for y (x) at every point of the range of integration [2,30].…”

Section: Introductionmentioning

confidence: 99%

Quadratic spline solution for boundary value problem of fractional order

Zahra

Elkholy

2011

Numer Algor

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Fractional differential equations are widely applied in physics, chemistry as well as engineering fields. Therefore, approximating the solution of differential equations of fractional order is necessary. We consider the quadratic polynomial spline function based method to find approximate solution for a class of boundary value problems of fractional order. We derive a consistency relation which can be used for computing approximation to the solution for this class of boundary value problems. Convergence analysis of the method is discussed. Four numerical examples are included to illustrate the practical usefulness of the proposed method.Keywords Boundary value problems of fractional order · Riemann-Liouville fractional derivative · Caputo fractional derivative · Quadratic polynomial spline · Error bound

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References

Micula

1999

Handbook of Splines

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A quartic C3-spline collocation method for solving second-order initial value problems

Cited by 17 publications

References 16 publications

Hierarchical cascade model leading to 7-th order initial value problem

Hierarchical cascade model leading to 7-th order initial value problem

Quadratic spline solution for boundary value problem of fractional order

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