We present a simple and effective Chebyshev polynomial scheme (CPS) combined with the method of fundamental solutions (MFS) and the equilibrated collocation Trefftz method for the numerical solutions ...of inhomogeneous elliptic partial differential equations (PDEs). In this paper, CPS is applied in a two-step approach. First, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then the resulting homogeneous equation is solved by boundary type methods including the MFS and the equilibrated collocation Trefftz method. Numerical results for problems on various irregular domains show that our proposed scheme is highly accurate and efficient.
Several meshless methods that are used to solve the partial differential equations are particular solutions based numerical methods. These numerical methods can only be applied to solve the partial ...differential equations if researchers have derived a particular solution of some equations beforehand. The main contribution of this article is the derivation of the family of particular solutions of the Poisson’s equation in 3D with the oscillatory radial basis functions in the forcing term. Numerical results obtained by solving three elliptic partial differential equations presented here validates the derived particular solutions in the method of particular solutions.
In this paper, we derive closed-form particular solutions of Matérn radial basis functions for the Laplace and biharmonic operator in 2D and Laplace operator in 3D. These derived particular solutions ...are essential for the implementation of the method of particular solutions for solving various types of partial differential equations. Four numerical examples in 2D and 3D are given to demonstrate the effectiveness of the derived particular solutions.
Particular solutions play a critical role in solving inhomogeneous problems using boundary methods such as boundary element methods or boundary meshless methods. In this short article, we derive the ...closed-form particular solutions for the Laplace and biharmonic operators using the Gaussian radial basis function. The derived particular solutions are implemented numerically to solve boundary value problems using the method of particular solutions and the localized method of approximate particular solutions. Two examples in 2D and 3D are given to show the effectiveness of the derived particular solutions.
Twisted laminar superconducting composite structures based on multi-wall carbon nanotube (MWCNT) yarns were crafted by integrating magnesium and boron homogeneous mixture into the carbon nanotube ...(CNT) aerogel sheets. After the ignition of the Mg–B–MWCNT system, under the controlled argon environment, the high exothermic reaction between magnesium (Mg) and boron (B) with stoichiometric ratio produced the MgB
2
@MWCNT superconducting composite yarns. The process was conducted under the controlled argon environment and uniform heating rate in the differential scanning calorimetry and thermogravimetric analyzer. The XRD analysis confirmed that the produced composite yarns contain nano and microscale inclusions of superconducting phase of MgB
2
. The mechanical properties of the composite twisted and coiled yarns at room temperature were characterized. The tensile strength up to 200 MPa and Young’s modulus of 1.27 GPa proved that MgB
2
@MWCNT composite is much stiffer than single component MgB
2
wires. The superconductive critical temperature of
T
c
~38 K was determined by measuring temperature-dependent magnetization curves. The critical current density,
J
c
of superconducting component of composite yarns was obtained at different temperatures below
T
c
by using magnetic hysteresis measurement. The highest value of
J
c
= 3.39 × 10
7
A cm
−2
was recorded at 5 K.
In recent years, localized methods are proven to be very effective for solving various types of problems in scientific computing. Many researchers have successfully implemented localized approaches ...to solve large-scale problems. Oscillatory radial basis functions collocation method, a global method, is a meshless numerical method in the literature. The novelty of this article is to address the computational efficiency issues of the oscillatory radial basis functions collocation method using a localized approach for solving elliptic partial differential equations in 2D. We carry out a number of experiments to validate our proposed numerical scheme. Numerical results clearly demonstrate that our scheme is highly accurate and computationally efficient.
The fast method of approximate particular solutions (FMAPS) is based on the global version of the method of approximate particular solutions (MAPS). In this method, given partial differential ...equations are discretized by the usual MAPS and the determination of the unknown coefficients is accelerated using a fast technique. Numerical results confirm the efficiency of the proposed technique for the PDEs with a large number of computational points in both two and three dimensions.
Despite all the efforts and success for finding the optimal location of the sources outside the domain for the method of fundamental solutions (MFS), this issue continues to attract the attention ...from researchers for seeking more efficient and reliable algorithms. In this paper, we propose to extend the adaptive greedy technique which applies the primal-dual formulation for the selection of source nodes in the MFS for Laplace equation with nonharmonic boundary conditions. Such approach is a data-dependent algorithm which adaptively selects the suitable source nodes based on the specific adaptive procedure. Both 2D and 3D examples are provided. Moreover, the proposed algorithm is easy to implement with high accuracy.
In this paper, we propose hybrid Chebyshev polynomial scheme (HCPS), which couples the Chebyshev polynomial scheme and the method of fundamental solutions into a single matrix system. This hybrid ...formulation requires solving only one system of equations and opens up the possibilities for solving a large class of partial differential equations. In this paper, we consider various boundary value problems and, in particular, the challenging Cauchy–Navier equation. The solution is approximated by the sum of the particular solution and the homogeneous solution. Chebyshev polynomials are used to approximate a particular solution of the given partial differential equation and the method of fundamental solutions is used to approximate the homogeneous solution. Numerical results show that our proposed approach is efficient, accurate, and stable.
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