Linearized Transformed $L1$ Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schrödinger Equations

Yuan, Wanqiu; null, Dongfang Li; Zhang, Chengjian

doi:10.4208/nmtma.oa-2022-0087

Cited by 16 publications

(2 citation statements)

References 39 publications

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“…The interpolation Π 1,γ N is identical to employing the basis function {1, t γ } to approximate the solution u to the considered fractional differential equation. This idea has been adopted to solve the time-fractional Black-Scholes equation [30], the multi-term time-fractional diffusion equation [29], and the nonlinear time-fractional Schrödinger equation [39].…”

Section: The L1 Methods and Its Variantsmentioning

confidence: 99%

L1 Schemes for time-fractional differential equations: A brief survey and new development

Xiao,

Chen,

Zeng

et al. 2024

Preprint

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We summarize the L1 method and its variants for the discretization of the Caputo derivative and develop a weighted L1 (WL1) method. The WL1 method includes the classical L1 method as a special case but improves the L1 method in accuracy while accommodating the intrinsic weak singularity of solutions to fractional differential equations. The WL1 method inherits several properties of the L1 method and the analysis of the new method is presented in a simple way. Compared to several variants of the L1 method, the new method accommodates the intrinsic weak singularity of the time-fractional derivatives in a more flexible way, especially for variable-order fractional differential equation. We also develop the fast WL1 method and apply it to initial and boundary value problems. Numerical simulations and comparisons verify our theoretical analysis and demonstrate the flexibility and efficiency of our method. Mathematics Subject Classification (2020) 26A33 · 65M06 · 65M12 · 65M15 · 35R11

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Section: The L1 Methods and Its Variantsmentioning

confidence: 99%

L1 Schemes for time-fractional differential equations: A brief survey and new development

Xiao,

Chen,

Zeng

et al. 2024

Preprint

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“…These problems involve summing Caputo derivatives with orders ranging from 0 to 1. A new linearized transformed L 1 Galerkin finite element approach is presented for numerical solutions of the multi-dimensional time fractional Schrödinger equations [45]. The fully discrete scheme's conditionally optimum error estimates are demonstrated.…”

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confidence: 99%

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

Nawaz,

Arif,

Mansoor

et al. 2024

Fractal Fract

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An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid’s fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme.

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Two‐grid finite element method on grade meshes for time‐fractional nonlinear Schrödinger equation

Hu,

Chen,

Zhou

2023

Numerical Methods Partial

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A two‐grid finite element method with nonuniform L1 scheme is developed for solving the time‐fractional nonlinear Schrödinger equation. The finite element solution in the ‐norm and ‐norm are proved bounded without any time‐step size conditions (dependent on spatial‐step size). Then, the optimal order error estimations of the two‐grid solution in the ‐norm are proved without any time‐step size conditions. Finally, the theoretical results are verified by numerical experiments.

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Linearized Transformed $L1$ Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schrödinger Equations

Cited by 16 publications

References 39 publications

L1 Schemes for time-fractional differential equations: A brief survey and new development

L1 Schemes for time-fractional differential equations: A brief survey and new development

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

Two‐grid finite element method on grade meshes for time‐fractional nonlinear Schrödinger equation

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