Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator

Płociniczak, Łukasz; Sobieszek, Szymon

doi:10.1007/s11075-016-0247-z

Cited by 9 publications

(6 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. The asymptotic relation for Erdélyi-Kober fractional integral operator I 0,β α/β was delivered in [37], therefore, we will not provide a detailed proof of it here.…”

Section: Discretization Of the Erdélyi-kober Differential Operatormentioning

confidence: 99%

“…It is worth to mention that in [37] authors proposed different discretization methods of the Erdélyi-Kober fractional integral operator. In addition to the rectangle rule they used also midpoint and trapezoid rule to obtain more accurate approximations.…”

Section: Discretization Of the Erdélyi-kober Differential Operatormentioning

confidence: 99%

“…Other operators are analysed analogously. This method of estimating the order of convergence was also used in [37] where different methods of discretization of the Erdélyi-Kober fractional integral operator were investigated. Here, as a test function we use φ(t) = e t and evaluate the error at t = t n = 1.…”

Section: Numerical Analysismentioning

confidence: 99%

See 2 more Smart Citations

Numerical scheme for Erdélyi-Kober fractional diffusion equation using Galerkin-Hermite method

Płociniczak¹,

Świtała²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erdélyi-Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion and grey Brownian motion. To obtain convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erdélyi-Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singular in time term appearing in the main equation the proposed method converges slower than first order. Finally, we provide the numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.

show abstract

“…Proof. The asymptotic relation for Erdélyi-Kober fractional integral operator I 0,β α/β was delivered in [37], therefore, we will not provide a detailed proof of it here.…”

Section: Discretization Of the Erdélyi-kober Differential Operatormentioning

confidence: 99%

Section: Discretization Of the Erdélyi-kober Differential Operatormentioning

confidence: 99%

See 1 more Smart Citation

Numerical scheme for Erdélyi-Kober fractional diffusion equation using Galerkin-Hermite method

Płociniczak¹,

Świtała²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Using formula (92) from [23] we can numerically check the order of convergence of the numerical scheme. The comparison prepared for both variables represent Table 1 and Table 2.…”

Section: Numerical Examples Example 1 Let Us Take Parametersmentioning

confidence: 99%

A weighted finite difference method for subdiffusive Black Scholes Model

Krzyżanowski,

Magdziarz,

Płociniczak

2019

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper we focus on the subdiffusive Black Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank-Nicolson scheme. The proposed method has 2 − α order of accuracy with respect to time where α ∈ (0, 1) is the subdiffusion parameter, and 2 with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some numerical results.

show abstract

“…Recall that for a smooth function, its Newton-Cotes quadrature has an error bounded by the value of a sufficiently high derivative. If the considered function is not smooth enough, the order of the method can be reduced [35,14]. On the other hand, when 0 < m < 1 the Corollary 1 can give very weak estimates on the convergence rate.…”

Section: Construction Of a Quadrature And Numerical Examplesmentioning

confidence: 99%

Numerical method for Volterra equation with a power-type nonlinearity

Okrasińska-Płociniczak

Płociniczak

2018

Applied Mathematics and Computation

Self Cite

View full text Add to dashboard Cite

In this work we prove that a family of explicit numerical finite-difference methods is convergent when applied to a nonlinear Volterra equation with a power-type nonlinearity. In that case the kernel is not of Lipschitz type, therefore the classical analysis cannot be applied. We indicate several difficulties that arise in the proofs and show how they can be remedied. The tools that we use consist of variations on discreet Gronwall's lemmas and comparison theorems. Additionally, we give an upper bound on the convergence order. We conclude the paper with a construction of a convergent method and apply it for solving some examples.

show abstract

Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator

Cited by 9 publications

References 54 publications

Numerical scheme for Erdélyi-Kober fractional diffusion equation using Galerkin-Hermite method

Numerical scheme for Erdélyi-Kober fractional diffusion equation using Galerkin-Hermite method

A weighted finite difference method for subdiffusive Black Scholes Model

Numerical method for Volterra equation with a power-type nonlinearity

Contact Info

Product

Resources

About