Numerical solution of nonlinear fractional differential equations by spline collocation methods

Pedas, Arvet; Tamme, Enn

doi:10.1016/j.cam.2013.04.049

Cited by 83 publications

(43 citation statements)

References 15 publications

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“…Applications of differentiation and integration with non-integer orders can be traced back to premature in history, so it can be said that it is not new [16]. Many different techniques and methods of dealing with fractional differential equations resulting analytical and numerical solutions can be found in a wide variety of studies in the literature [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of time fractional Burgers equation

Esen

Taşbozan

2015

Acta Universitatis Sapientiae, Mathematica

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Section: Introductionmentioning

confidence: 99%

Numerical solution of time fractional Burgers equation

Esen

Taşbozan

2015

Acta Universitatis Sapientiae, Mathematica

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“…There are several numerical methods for solving fractional differential equations, such as Haar wavelet method [5], spline collocation method [6], fractional differential transform method [7].…”

Section: Introductionmentioning

confidence: 99%

National economies in state-space of fractional-order financial system

et al. 2015

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“…Therefore, we have to apply some numerical methods for solving (1.1). Stability and convergence of such numerical methods are analyzed under certain smoothness assumptions for the solutions of (1.1); see, for example, [1,2,7,11,14,17,18,23,26].…”

Section: Introductionmentioning

confidence: 99%

Detailed error analysis for a fractional Adams method with graded meshes

2017

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We consider a fractional Adams method for solving the nonlinear fractional differential equation C 0 D α t y(t) = f (t, y(t)), α > 0, equipped with the initial conditions y (k) Here, α may be an arbitrary positive number and α denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption 2004) introduced a fractional Adams method with the uniform meshes t n = T (n/N), n = 0, 1, 2, . . . , N and proved that this method has the optimal convergence order uniformly in t n , that is, the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y ∈ C m [0, T ] for some m ∈ N and 0 < α < m, the Caputo fractional derivative C 0 D α t y(t) takes the form " C 0 D α t y(t) = ct α −α + smoother terms" (Diethelm et al. Numer. Algor. 36, 2004), which implies that C 0 D α t y behaves as t α −α which is not in C 2 [0, T ]. By using the graded meshes t n = T (n/N) r , n = 0, 1, 2, . . . , N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in t n even if C 0 D α t y behaves as

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Numerical solution of nonlinear fractional differential equations by spline collocation methods

Cited by 83 publications

References 15 publications

Numerical solution of time fractional Burgers equation

Numerical solution of time fractional Burgers equation

National economies in state-space of fractional-order financial system

Detailed error analysis for a fractional Adams method with graded meshes

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