scholarly journals Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 394 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Muhammad Ahsan ◽  
Iltaf Hussain ◽  
Poom Kumam ◽  
Wiyada Kumam
Keyword(s):  
Numerical Simulation ◽  
Differential Equations ◽  
Collocation Method ◽  
Wave Equations ◽  
Differential Quadrature ◽  
Test Problems ◽  
Difference Formula ◽  
Meshless Collocation ◽  
Klein Gordon ◽  
Convection Dominated

In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems.

scholarly journals Numerical Simulation of Partial Differential Equations via Local Meshless Method

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 257 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Muhammad Riaz ◽  
Muhammad Ayaz ◽  
Muhammad Arif ◽  
Saeed Islam ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Meshless Method ◽  
Three Dimensional ◽  
Salient Feature ◽  
Test Problems ◽  
Spatial Discretization ◽  
Partial Differential ◽  
Meshless Collocation

In this paper, numerical simulation of one, two and three dimensional partial differential equations (PDEs) are obtained by local meshless method using radial basis functions (RBFs). Both local and global meshless collocation procedures are used for spatial discretization, which convert the given PDEs into a system of ODEs. Multiquadric, Gaussian and inverse quadratic RBFs are used for spatial discretization. The obtained system of ODEs has been solved by different time integrators. The salient feature of the local meshless method (LMM) is that it does not require mesh in the problem domain and also far less sensitive to the variation of shape parameter as compared to the global meshless method (GMM). Both rectangular and non rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy and ease implementation of the proposed method are shown via test problems.

scholarly journals A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1336
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Dumitru Ţucu ◽  
Marioara Lăpădat ◽  
Mădălina Sofia Paşca
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Least Squares ◽  
Least Squares Method ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Differential Quadrature ◽  
Test Problems ◽  
Quadrature Method ◽  
Partial Differential

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

A new structure formulations for cubic B-spline collocation method in three and four-dimensions

2020 ◽  
Vol 9 (1) ◽  
pp. 432-448
Author(s):  
K. R. Raslan ◽  
Khalid K. Ali
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Collocation Method ◽  
Test Problems ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method ◽  
Four Dimensions ◽  
B Splines

AbstractIn this work, we introduce a new construct to the cubic B-spline collocation method in the three and four-dimensions. The cubic B-splines method format is displayed in one, two, three, and four-dimensions format. These constructions are of utmost importance in solving differential equations in their various dimensions, which have applications in many fields of science. The efficiency and accuracy of the proposed methods are demonstrated by its application to a few test problems in two, three, and four dimensions. Also, comparing the exact solutions and with the results obtained by using other numerical methods available in the literature as much as possible.

An Adaptive Wavelet Collocation Method for Solution of the Convection-Dominated Problem on a Sphere

2018 ◽  
Vol 15 (08) ◽  
pp. 1850080
Author(s):  
Ratikanta Behera ◽  
Mani Mehra
Keyword(s):  
Numerical Simulation ◽  
Collocation Method ◽  
Second Generation ◽  
Efficient Computation ◽  
Adaptive Wavelet ◽  
Spherical Wavelets ◽  
Collocation Points ◽  
Wavelet Collocation Method ◽  
Uniform Grids ◽  
Convection Dominated

In this paper, we present a numerical simulation of a convection-dominated problem using adaptive wavelet collocation method. This is based on second-generation spherical wavelets on a recursively refined spherical geodesic grid. The power lies in the fact that they only require small number of coefficients to represent this problem more accurately, which allows compression and efficient computation. In contrast with other approximate schemes, this approach reduced the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution. Numerical results demonstrate the efficiency of our approach.

Numerical solution of parabolic partial differential equations using adaptive gird Haar wavelet collocation method

2016 ◽  
Vol 10 (02) ◽  
pp. 1750026 ◽  
Author(s):  
S. C. Shiralashetti ◽  
L. M. Angadi ◽  
M. H. Kantli ◽  
A. B. Deshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Haar Wavelet ◽  
Parabolic Partial Differential Equations ◽  
Test Problems ◽  
Parabolic Pdes ◽  
Partial Differential ◽  
Wavelet Collocation Method

In this paper, we applied the adaptive grid Haar wavelet collocation method (AGHWCM) for the numerical solution of parabolic partial differential equations (PDEs). The approach of AGHWCM for the numerical solution of parabolic PDEs is mentioned, the obtained numerical results, error analysis are presented in figures and tables. This shows that, the AGHWCM gives better accuracy than the HWCM and FDM. Some of the test problems are taken for demonstrating the validity and applicability of the AGHWCM.

Relativistic Wave Equations

2021 ◽  
pp. 167-190
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Wave Equations ◽  
Plane Waves ◽  
Canonical Formalism ◽  
Real Field ◽  
Klein Gordon ◽  
Linear Differential ◽  
Propagation Properties

We derive the most general relativistically covariant linear differential equations, having at most two derivatives, for scalar, spinor and vector fields. We introduce the corresponding Lagrangian and Hamiltonian formalisms and present the expansion of the solutions in terms of plane waves. In each case, we study the propagation properties of the corresponding Green functions. We start with the simplest example of the Klein–Gordon equation for a real field and generalise it to that of N real, or complex fields. As a next step we derive the Weyl, Majorana and Dirac equations for spinor fields. They are first order differential equations and we show how to adapt to them the canonical formalism. We end with the Proca and Maxwell equations for massive and massless spin-one fields and, in each case, we determine the physical degrees of freedom.

scholarly journals Numerical simulation for fractional-order differential system of a Glioblastoma Multiforme and Immune system

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
M. M. Al-Shomrani ◽  
M. A. Abdelkawy
Keyword(s):  
Numerical Simulation ◽  
Immune System ◽  
Glioblastoma Multiforme ◽  
Collocation Method ◽  
Fractional Order ◽  
Differential System ◽  
Test Problems ◽  
The Novel ◽  
Orthogonal Functions ◽  
Novel Method

Abstract In this paper, we present a numerical simulation to study a fractional-order differential system of a glioblastoma multiforme and immune system. This numerical simulation is based on spectral collocation method for tackling the fractional-order differential system of a glioblastoma multiforme and immune system. We introduce new shifted fractional-order Legendre orthogonal functions outputted by Legendre polynomials. Also, we state and derive some corollaries and theorems related to the new shifted fractional order Legendre orthogonal functions. The shifted fractional-order Legendre–Gauss–Radau collocation method is developed to approximate the fractional-order differential system of a glioblastoma multiforme and immune system. The basis of the shifted fractional-order Legendre orthogonal functions is adapted for temporal discretization. The solution of such an equation is approximated as a truncated series of shifted fractional-order Legendre orthogonal functions for temporal variable, and then we evaluate the residuals of the mentioned problem at the shifted fractionalorder Legendre–Gauss–Radau quadrature points. The accuracy of the novel method is demonstrated with several test problems.

A localized Newton basis functions meshless method for the numerical solution of the 2D nonlinear coupled Burgers’ equations

2017 ◽  
Vol 27 (11) ◽  
pp. 2582-2602 ◽  
Author(s):  
Fahimeh Saberi Zafarghandi ◽  
Maryam Mohammadi ◽  
Esmail Babolian ◽  
Shahnam Javadi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Meshless Method ◽  
Basis Functions ◽  
Content Type ◽  
Burgers Equations ◽  
Convection Dominated ◽  
Newton Basis Functions

Purpose The purpose of this paper is to introduce a local Newton basis functions collocation method for solving the 2D nonlinear coupled Burgers’ equations. It needs less computer storage and flops than the usual global radial basis functions collocation method and also stabilizes the numerical solutions of the convection-dominated equations by using the Newton basis functions. Design/methodology/approach A meshless method based on spatial trial space spanned by the local Newton basis functions in the “native” Hilbert space of the reproducing kernel is presented. With the selected local sub-clusters of domain nodes, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs). Findings The method is successfully used for solving the 2D nonlinear coupled Burgers’ equations for reasonably high values of Reynolds number (Re). It is a well-known issue in the analysis of the convection-diffusion problems that the solution becomes oscillatory when the problem becomes convection-dominated if the standard methods are followed without special treatments. In the proposed method, the authors do not detect any instability near the front, hence no technique is needed. The numerical results show that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re. Originality/value The authors used more stable basis functions than the standard basis of translated kernels for representing of kernel-based approximants for the numerical solution of partial differential equations (PDEs). The local character of the method, having a well-structured implementation including enforcing the Dirichlet and Neuman boundary conditions, and producing accurate and stable results for flow with reasonably high values of Re for the numerical solution of the 2D nonlinear coupled Burgers’ equations without any special technique are the main values of the paper.

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Miaomiao Yang ◽  
Xinkun Du ◽  
Yongbin Ge
Keyword(s):  
Differential Equations ◽  
Helmholtz Equation ◽  
Collocation Method ◽  
Meshless Method ◽  
Lagrange Interpolation ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Wave Number ◽  
Content Type ◽  
Meshless Collocation

PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

Sign in / Sign up

Export Citation Format

Share Document