An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

中文翻译：

---

用于时间分数非线性弱奇异积分-偏微分方程的同步空-时 Hermite 小波方法


已经开发了一种创新的空-时同步 Hermite 小波方法，用于求解一维和二维中的弱奇异分数阶非线性积分偏微分方程，其重点是其解在空间和时间上都是间歇性的。所提出的方法基于多维 Hermite 小波和准线性化技术。联立时空方法没有充分利用时间分数非线性弱奇异积分偏微分方程。随后，当解取决于整个时域（包括过去和未来时间）时，收敛分析具有挑战性，并且控制方程与 Volterra 和 Fredholm 积分算子相结合。考虑到这些挑战，我们使用准线性化技术来处理问题的非线性，并将其重建为具有二阶精度的线性积分-偏微分方程。然后，我们将多维 Hermite 小波作为有吸引力的候选者应用于得到的线性化问题，以有效解决 t=0 时的初始弱奇点。此外，搭配方法用于确定分解域内基于张量的小波系数。我们详细阐述了所提出的同步时空 Hermite 小波方法的构建，并为其实现设计了全面的算法。具体来说，我们强调 L2 范数框架内的收敛分析，并表明高精度取决于解的规律性。 所提出的基于小波的数值近似的稳定性也在分数阶非线性积分偏微分方程的背景下进行了讨论，该方程涉及具有弱奇异核的 Volterra 和 Fredholm 算子。将所提出的方法与文献中可用的现有方法进行了比较。具体来说，我们强调了它的高精度，并将其与最近开发的混合数值方法和有限差分方法进行了比较。通过求解几个高度间歇的时间分数非线性弱奇异积分偏微分方程，证明了所提方法的效率和准确性。