Point interpolation collocation method for the solution of partial differential equations

In this article, recently proposed spectral meshless radial point interpolation (SMRPI) method is applied to the two-dimensional diffusion equation with a mixed group of Dirichlet's and Neumann's and non-classical boundary conditions. The present method is based on meshless methods and benefits from spectral collocation ideas. The point interpolation method with the help of radial basis functions is proposed to construct shape functions which have Kronecker delta function property. Evaluation of high-order derivatives is possible by constructing and using operational matrices. The computational cost of the method is modest due to using strong form equation and collocation approach. A comparison study of the efficiency and accuracy of the present method and other meshless methods is given by applying on mentioned diffusion equation. Stability and convergence of this meshless approach are discussed and theoretically proven. Convergence studies in the numerical examples show that SMRPI method possesses excellent rates of convergence.

Dispersion analysis for acoustic problems using the point interpolation method

In this paper, a pseudospectral meshless radial point interpolation (PSMRPI) technique is applied to the three-dimensional wave equation with variable coefficients subject to given appropriate initial and Dirichlet boundary conditions. The present method is a kind of combination of meshless methods and spectral collocation techniques. The point interpolation method along with the radial basis functions is used to construct the shape functions as the basis functions in the frame of the spectral collocation methods. These basis functions will have Kronecker delta function property, as well as unitary possession. In the proposed method, operational matrices of higher order derivatives are constructed and then applied. The merit of this innovative method is that, it does not require any kind of integration locally or globally over sub-domains, as it is essential in meshless methods based on Galerkin weak forms, such as element-free Galerkin and meshless local Petrov–Galerkin methods. Therefore, computational cost of PSMRPI method is low. Further, it is proved that the procedure is stable with respect to the time variable over some conditions on the 3D wave model, and the convergence of the technique is revealed. These latest claims are also shown in the numerical examples, which demonstrate that PSMRPI provides excellent rate of convergence.

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

For the better management of available groundwater, it is essential to understand the flow behavior within the aquifer systems. The groundwater flow behavior in the complex aquifer system can be studied by solving the governing equations using either analytical or numerical methods. As the analytical solutions are available only for simple idealized cases, numerical methods such as finite difference method (FDM) and finite element method (FEM) are generally used. The Meshless method is a recently developed alternative numerical approach for the solution of existing problem. A variety of Meshless methods like point interpolation method, smooth particle hydrodynamics, Galerkin methods, etc. are under the development for the solution of many solid mechanics and fluid mechanics problems. Meshless method eliminates many of the drawbacks of grid-based methods such as difficulties in meshing and re-meshing as in FDM and FEM which require huge efforts in pre-processing and with Meshless method it translates in the lesser computational time and costs. In this paper, Meshless Local Petrov-Galerkin (MLPG) method with strong form of collocation with exponential/Gaussian radial basis function is used for solving the groundwater flow problem. A computer model in MATLAB has been developed in 1D and 2D for the solution of confined aquifer problems. The developed 1D model is verified with available analytical solution for a hypothetical problem and found to be satisfactory. Further, the 2D model is applied to two case studies and results are compared with the FEM based solutions and found to be satisfactory. The present study shows that the MLPG based Meshless method is very effective in the simulation of groundwater flow problems.

This paper develops pseudospectral meshless radial point Hermit interpolation (PSMRPHI) and pseudospectral meshless radial point interpolation (PSMRPI) in order to apply to the elliptic partial differential equations (PDEs) held on irregular domains subject to impedance (convective) boundary conditions. Elliptic PDEs in simplest form, i.e., Laplace equation or Poisson equation, play key role in almost all kinds of PDEs. On the other hand, impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems, are nearly more complicated forms of the boundary conditions in boundary value problems (BVPs). Based on this problem, we aim also to compare PSMRPHI and PSMRPI which belong to more influence type of meshless methods. PSMRPI method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of PSMRPI and PSMRPHI methods. While the latter one has been rarely used in applications, we observe that is more accurate and reliable than PSMRPI method.

In this chapter we discuss the prerequisites that are necessary to understand how problems in the natural sciences are formulated using (differential) equations and how to solve them numerically. For this, we discuss the basics of ordinary and partial differential equations in Sect. 4.1. We learn about the Lipschitz condition for ordinary differential equations and about initial and boundary conditions with partial differential equations, where we distinguish between three basic types of equations. A discussion of plane and spherical wave solutions of partial differential equations leads us to the study of numerical procedures in Sect. 4.2. Here, we discuss mesh-based and mesh-free methods, finite difference, finite volume, and finite element methods. We also get to know the most important algorithm used for numerical integration in molecular dynamics methods – the velocity Verlet algorithm. Finally, in Sect. 4.3 some tricks of the trade and important principles in software design and engineering which are not found in other books on computational science are presented. The chapter ends with a couple of problems for the interested reader.

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

This paper presents a finite element-least square point interpolation method (FE-LSPIM) for solving 2D acoustic problem by synthesizing the attractive features of finite element and least-square point interpolation shape functions using the concepts of partition of unity (PU) methods. In the present method, the acoustic domain is discretized using quadrilateral element, and the shape functions of the quadrilateral element are used for PU and the least-square point interpolation method (LSPIM) for local approximation. This enables the proposed method to inherit the compatibility properties of finite element method and the quadratic polynomial completeness properties of meshfree methods, so that the finite element-least-square point interpolation method (FE-LSPIM) will greatly reduce the numerical dispersion error because the numerical dispersion error is essentially caused by the �overly-stiff� nature of the FEM model. Numerical results for benchmark problems show that, the FE-LSPIM achieves more accurate results and higher convergence rates as compared with the corresponding finite elements and Element-free Galerkin method (EFGM), especially for high wave number and irregular meshes

A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional Fredholm integral equations on general domains with error analysis

The fractional partial differential equation is an equation with non-integer partial derivative which is now widely used in solving various problems so that in recent years, many researchers are interested to study fractional partial differential equation. The equation which is used in this paper is the non-homogeneous fractional partial differential equation. The aim of this study is to determine general solution of the partial differential equation through three different order equations (α; 1), (β; 2) and (α; β), where 0 < α ≤ 2 and 0 < β ≤ 1 by using Homotopy Analysis Method (HAM). Furthermore, by taking a convergent sequence, it will be analyzed that the convergence of the sequence of differential equations results in the sequence of solutions of partial fractional non-homogeneous partial differential equations converging to the solution function of the fractional partial differential equation.

Several research papers experimentally validated the derived theoretical solution of the partial differential equation (PDE) of Transient Land Drainage (TLD) using laboratory sand tanks, Hele-Shaw models, resistance networks, and field data. In this research, the main objective is to use an Arduino microcontroller linked with the LabVIEW software to derive experimentally-by using an RC analog circuit connected to the Arduino microcontroller - a solution to Kirchhoff’s partial differential equation describing the unsteady electrical flow through the RC circuit. The RC circuit electrically simulates a non-dimensional virtual aquifer for which an experimental Fourier sine series solution in terms of the analog components is to be derived and used for comparison with the theoretical Fourier sine series solution of the transient partial differential equation of land drainage. The derived Fourier sine series solution of the partial differential equation governing the unsteady electrical flow through the RC analog is written in a non-dimensional form to enable its general application to the case study of this research. Isochrones of the water table drawdown based on those derived from the analog solution are similar in trend and behavior to those of the theoretical solution, indicating that both Fourier solutions are physically and mathematically identical.

Functionally graded materials (FGMs) are complicated composites created using the concept of continuous variation of material property in one or more predetermined directions. FGMs have been used in the manufacturing of structural parts that are subjected to non-uniform functioning requirements in recent years. In a thermal protection system, for example, FGMs combine the benefits of traditional ceramics, such as corrosion and heat resistance, with those of metal, such as mechanical strength and rigidity. They are common in engineering practice, so static analysis and dynamic analysis for FGM plate structures are necessary. The radial point interpolation method (RPIM) meshless method has been used based on the point interpolation method (PIM) by including the radial basis function (RPF) in the interpolation formulation and has shown good performance in computational engineering. One of the advantages of this method is that it satisfies the Kronecker’s delta function, which overcomes the limitations of critical boundary conditions for the traditional meshless method. Furthermore, RPIM shape function satisfies the high-order continuity constraint, which the low-order finite element methods (FEM) method does not. This paper presents a meshless approach for the static and dynamic analysis of FGM plates whose material properties vary through the thickness. Numerical examples are solved and the results are compared with reference solutions or the results of FEM given by SOMSOL program to confirm the accuracy of the proposed method.

The main problem with simulating the partial differential equation (PDE) solution process using neural networks is the need to increase the accuracy and size of the domain. This paper proposes improving the convolutional neural network (CNN) architecture to simulate solving a boundary value problem based on the Poisson equation with arbitrary Dirichlet and Neumann boundary conditions, which reduces the time of calculations by 3.36 times with an increase in accuracy of calculations by 1.4 times. We achieve the obtained result by modifying the CNN’s architecture at the stages of data input and their further processing. The paper compares available approaches to modeling this problem’s solution and the solution’s results by the finite element method. We obtained the best results in both cases, showing relevant practical study examples.

Interface solutions of partial differential equations with point singularity