Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg-Landau equation

Zhang, Zuo‐Feng; Hesthaven, Jan S.; Sun, Zhi‐zhong; Ren, Yunzhu

doi:10.1007/s10444-021-09862-x

Cited by 19 publications

(3 citation statements)

References 53 publications

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“…Inserting the result of ( 25) into (24), then using it with (6), the bright soliton solutions for the commanding model (1) can be introduced as follows:…”

Section: Bright Solitonmentioning

confidence: 99%

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

et al. 2022

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In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well.

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“…Inserting the result of ( 25) into (24), then using it with (6), the bright soliton solutions for the commanding model (1) can be introduced as follows:…”

Section: Bright Solitonmentioning

confidence: 99%

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

et al. 2022

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“…Recently, the fractional models have aroused numerous research interests in various fields, ranging from finance, 28,29 neuroscience, 30,31 physics, 32,33 and so on. [34][35][36][37][38][39] We here focus on the fractional model in option pricing. The finite difference method is a widely used method in option pricing.…”

Section: Monte Carlo and Finite Difference Of Fpdementioning

confidence: 99%

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Yuan¹,

Ding²,

Duan³

et al. 2021

Preprint

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During the COVID-19 pandemic, many institutions have announced that their counterparties are struggling to fulfill contracts. Therefore, it is necessary to consider the counterparty default risk when pricing options. After the 2008 financial crisis, a variety of value adjustments have been emphasized in the financial industry. The total value adjustment (XVA) is the sum of multiple value adjustments, which is also investigated in many stochastic models such as Heston 1 and Bates 2 models. In this work, a widely used pure jump Lévy process, the CGMY process has been considered for pricing a Bermudan option with various value adjustments. Under a pure jump Lévy process, the value of derivatives satisfies a fractional partial differential equation (FPDE). Therefore, we construct a method which combines Monte Carlo with finite difference of FPDE (MC-FF) to find the numerical approximation of exposure, and compare it with the benchmark Monte Carlo-COS (MC-COS) method. We use the discrete energy estimate method, which is different with the existing works, to derive the convergence of the numerical scheme. Based on the numerical results, the XVA is computed by the financial exposure of the derivative value.

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“…Competitiveness of the numerical methods based on the analytical solution of ODEs is enhanced through the above results. However, numerical quadrature methods with matrix functions as integrands are rarely reported ( [19,20]).…”

Section: Introductionmentioning

confidence: 99%

Numerical quadrature methods for systems of linear ODEs with constant coefficients

Lin,

Yang,

Zhang

2024

Preprint

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In this paper, we consider high precision numerical methods for the initial problem of systems of linear ordinary differential equations (ODEs) with constant coefficients. It is well-known that the analytic solution (AS) of the initial value problem for a system of linear ODEs with constant coefficients involves a matrix exponential function, and an integral whose inte-grand is the product of a matrix exponential function and a vector-valued function. We mainly consider numerical quadrature methods for the integral term in the analytic solution. To simplify the discussion, we consider the case where the coefficient matrix is real and symmetric, and compute the corresponding matrix exponential function by using matrix spectral factorization. We propose a product quadrature (PQ) method, including its detailed computational procedure and its convergence analysis. Then the proposed methods are applied to the initial-boundary value problem for a heat conduction equation and a Riesz space fractional diffusion equation, respectively. Applying the central finite difference operator to discretize the second-order space derivative or a weighted shifted Grünwald difference (WSGD) operator to discretize the Riesz space fractional derivative, the initial-boundary value problem for the above mentioned equations is transformed to the initial value problem of a system of linear ODEs with constant coefficients. We then transform the systems of linear ODEs to symmetric ones if necessary. Numerical results presented to demonstrate the effectiveness of the proposed methods. Mathematics Subject Classification (2010) MSC 65M06 · MSC 26A33 · MSC 65M12

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Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg-Landau equation

Cited by 19 publications

References 53 publications

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Numerical quadrature methods for systems of linear ODEs with constant coefficients

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