Abstract
In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered, highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm.
Highlights
In recent decades, scientists and researchers have paid much attention to the expression of the physical models and chemical processes in the form of fractional derivative equations
In this work for solving the wave and damped wave equation with Riemann–Liouville fractional derivatives, we focus on the space–time Gaussian radial basis function in a high-dimensional setting
The proposed technique has some noticeable features, such as constructing a sparse matrix that reduces the time execution of the implementation of algorithms. This feature allows the method to be applied to highdimensional problems that occur in nature and engineering as well as the using the radial basis functions (RBFs) makes it much easier to work on high-dimensional spaces with irregular computational domains
Summary
Scientists and researchers have paid much attention to the expression of the physical models and chemical processes in the form of fractional derivative equations. The development of fractional calculus theory has provided an advantageous tool for modelling many natural processes that often have complex and anomalous modeling and cannot be expressed with classical derivative calculus. From about 20 years, radial basis functions (RBFs) are a primary mesh-free method for numerically solving PDEs on with collocation approaches (see, for example [3,4,5]). The numerical solution of partial differential equations by a global collocation approach based on RBF, is referred to as a strong form solution in the PDE literature [7,8,9,10]. An alternative interesting approach in collocation methods is to use other bases as for example Hermite exponential spline defined in [11]
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