Abstract In this paper, we construct a novel linearized finite difference/spectral-Galerkin scheme for three-dimensional distributed-order time–space fractional nonlinear reaction–diffusion-wave equation. By using Gauss–Legendre quadrature rule to discretize the distributed integral terms in both the spatial and temporal directions, we first approximate the original distributed-order fractional problem by the multi-term time–space fractional differential equation. Then, we employ the finite difference method for the discretization of the multi-term Caputo fractional derivatives and apply the Legendre–Galerkin spectral method for the spatial approximation. The main advantage of the proposed scheme is that the implementation of the iterative method is avoided for the nonlinear term in the fractional problem. Additionally, numerical experiments are conducted to validate the accuracy and stability of the scheme. Our approach is show-cased by solving several three-dimensional Gordon-type models of practical interest, including the fractional versions of sine-, sinh-, and Klein–Gordon equations, together with the numerical simulations of the collisions of the Gordon-type solitons. The simulation results can provide a deeper understanding of the complicated nonlinear behaviors of the 3D Gordon-type solitons.

中文翻译：

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三维分布阶次时空分数阶非线性反应扩散波方程的线性化有限差分/谱伽辽金格式：戈登型孤子的数值模拟

摘要 在本文中，我们为三维分布阶时空分数阶非线性反应扩散波方程构造了一种新的线性化有限差分/谱伽辽金格式。通过使用 Gauss-Legendre 求积法则在空间和时间方向上离散分布式积分项，我们首先通过多项时空分数阶微分方程来近似原始分布阶分数问题。然后，我们采用有限差分方法对多项 Caputo 分数阶导数进行离散化，并应用 Legendre-Galerkin 谱方法进行空间逼近。所提出方案的主要优点是避免了对分数问题中非线性项的迭代方法的实现。此外，数值实验验证了该方案的准确性和稳定性。我们的方法通过求解几个具有实际意义的三维 Gordon 型模型来展示，包括正弦、sinh 和 Klein-Gordon 方程的分数版本，以及 Gordon 型碰撞的数值模拟孤子。仿真结果可以更深入地了解 3D Gordon 型孤子的复杂非线性行为。