In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving an integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunfor...

Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

ESG scores and target price accuracy: Evidence from sell-side recommendations in BRICS

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N\log_2N)^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time-discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schr{o}dinger equation by using the fourth-order time-splitting method together with the proposed MCF-spectral-Galerkin method.

It is essential to explore the asymptotic stability of the numerical methods for fractional differential equations with delay since their exact solutions are unavailable or difficult to obtain. In this paper, we firstly derive a sufficient condition of the exact solution to be delay-independently asymptotically stable for the space fractional generalized diffusion equation with delay. Next, by applying the linear $$\theta $$-method to temporal dimension, we have established two classes of finite difference schemes independent of the delay for that, which have second-order and fourth-order accuracy in spatial dimension respectively. The unique solvability is proved and the local truncation error is derived. Furthermore, when $$\theta \in [0,1/2)$$, we have obtained the necessary and sufficient condition of the asymptotic stability of both schemes under the specific conditions for the temporal step size $$\Delta t$$ and the spatial step size h. When $$\theta \in [1/2,1]$$, both schemes are unconditionally asymptotically stable. Moreover, the convergence results in the maximum norm are obtained according to the consistency analysis and the Lax theorem. The numerical examples are implemented to verify the theoretical results, which demonstrate that the asymptotic stabilities of both schemes are depending on the parameter $$\theta $$ and the fractional derivative $$\alpha $$, however, independent of the delay $$\tau $$.

Asymptotic Stability of Compact and Linear $$\theta $$θ-Methods for Space Fractional Delay Generalized Diffusion Equation

Using the weighted and shifted Lubich difference (WSLD) operator, we propose an efficient and stable difference scheme for solving space tempered fractional diffusion equations. The scheme has four...

A High Order Finite Difference Method for Tempered Fractional Diffusion Equations with Applications to the CGMY Model

Analyst target price revisions and institutional herding

• We improve the objective function of the LSM and obtain the optimal SGFD coefficients for a flexible range of wave numbers. • A new optimal ESGFD and ISGFD scheme based on ANNs architecture is proposed. • The SGFD-ANNs method is applied to wave equation modeling, numerical experiments show the utility of the method. High-order staggered-grid finite-difference methods have been widely used in wave equation modeling. Imprecise difference coefficients and inappropriate difference order may result in numerical dispersion. We extend least square method (LSM) objective function to a flexible wave band to obtain its difference coefficients. Our method (model-driven LSM) is based on flexible model parameters to obtain corresponding optimal difference coefficients, rather than a uniform set of coefficients applied to all models. To reduce the computation cost in parameter estimation, we introduce the architecture, training and application of artificial neural networks (ANNs) to evaluate the staggered-grid finite-difference order and coefficients efficiently. In ANNs, double loss functions are adopted to ensure the stable and efficient convergence of training. Several numerical experiments for both two-dimensional and three-dimensional wave modeling are provided to verify the efficiency of the proposed scheme. 

Optimal staggered-grid finite-difference method for wave modeling based on artificial neural networks

Variable-order time-fractional diffusion equation with Mittag-Leffler kernel: regularity analysis and uniqueness of determining variable order

We concentrate on the analytical study on the pricing model for options written on multi-assets. These assets are assumed to follow a multi-variate CGMYe process through linear combination. We show that a European option satisfies a tempered fractional partial differential equation, for which the fundamental solution is given in an explicit form. For American options, the decomposition formula and the integral equation for the optimal-exercise boundary are also established. Furthermore, parameter estimations are implemented for the multi-dimensional problem, and several numerical simulations are provided for a European basket put.

Tempered fractional diffusion equations for pricing multi-asset options under CGMYe process

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