Global Collocation Method Research Articles

Singular expansions and collocation methods for generalized Abel integral equations

In this paper, the generalized Abel integral equation with singular solution is studied. First, the finite-term psi-series expansion of the solution about the origin is derived using the method of undetermined coefficients, which gives the complete singular information of the solution. Second, this singular expansion is used to separate the singularity such that the Abel integral equation is converted into a perturbed one with smooth solution on a regular interval. Third, a product trapezoidal method is designed to solve the transformed equation on the regular interval and the convergence analysis is conducted to show that the scheme has second order accuracy combining with a controllable perturbation term. Fourth, a Chebyshev collocation method is further constructed on the regular interval to show that the global orthogonal polynomial interpolation is also suitable for solving the transformed Abel integral equation with high accuracy. Finally, two numerical examples confirm the correctness of the truncated psi-series solution and the effectiveness of the piecewise and global collocation methods with singularity separation for solving this kind of Abel integral equation.

Soliton wave solutions of nonlinear mathematical models in elastic rods and bistable surfaces

The nonlinear wave phenomenon constitutes a significant research field and is a capable mathematical model for representing the transmission of energy in physical processes. This paper concentrates on the numerical solution for the nonlinear regularized long-wave and nonlinear extended Fisher–Kolmogorov models in two-space dimension. The solutions of these models are approximated via the finite difference technique and the localized radial basis function partition of unity. The association of the technique results in a meshfree method that does not require linearizing the nonlinear terms. In the first step, the partial differential equation (PDE) is transformed to a system of nonlinear ordinary differential equations (ODEs) by means of radial kernels. Subsequently, the nonlinear ODE system is approximated using a high-order ODE solver. The global collocation methods pose a considerable computational burden due to the calculation of the dense algebraic system. The proposed approach is based on a decomposition of the original domain into several subdomains via a kernel approximation on every subdomain. Numerical results such as the motion of single solitary and two solitary waves confirm the efficiency and accuracy of the method and are good agreement with the results of existing works in literature.

Coupling of the Crank–Nicolson scheme and localized meshless technique for viscoelastic wave model in fluid flow

This paper proposes an efficient localized meshless technique for approximating the viscoelastic wave model. This model is a significant methodology to explain wave propagation in solids modeled with a wide collection of viscoelastic laws. In the first method, a difference scheme with the second-order accuracy is implemented to obtain a semi-discrete scheme. Then, a localized radial basis function partition of unity scheme is adopted to get a full-discrete scheme. This localization technique consists of decomposing the initial domain into several sub-domains and constructing a local radial basis function approximation over every sub-domain. A well-conditioned resulting linear system and a low computational burden are the main merits of this technique compared to global collocation methods. Further, the stability and convergence analysis of the temporal discretization scheme are deduced using discrete energy method. Numerical results are shown to validate the accuracy and effectiveness of the proposed method.

An improved localized radial basis-pseudospectral method for solving fractional reaction–subdiffusion problem

The fractional reaction–subdiffusion problem is one of the most well-known subdiffusion models extensively used for simulating numerous physical processes in recent years. This paper introduces an efficient local hybrid kernel meshless procedure to approximate the time fractional reaction–subdiffusion problem involving a Riemann–Liouville fractional derivative. This technique is based on a central difference approach in the temporal direction and the hybridization of the cubic and Gaussian kernels in the spatial direction. The main idea of this hybridization is to develop a kernel that benefits from the advantages of two different kernels and avoids their limitations, while maintaining the global collocation method. This local approach considers only neighboring collocation nodes and it does not produces ill-conditioning that occurs in other methods with large dense matrix systems. The time discrete scheme in terms of the unconditional stability and convergence is analyzed. Numerical examples are presented to show the validity, effectiveness and accuracy of the method.

The impact of LRBF-FD on the solutions of the nonlinear regularized long wave equation

This paper develops a method for the numerical solution of the nonlinear regularized long wave equation. This method discretizes the unknown solution in two main schemes. The time discretization is accomplished by means of an implicit method based on the $$\theta $$ -weighted and finite difference methods, while the spatial discretization is described with the help of the finite difference scheme derived from the local radial basis function method. The advantage of the local collocation method is based only the discretization nodes located in each sub-domain, requiring to be considered when obtaining the approximate solution at every node. It also tackles the ill-conditioning problem derived from global collocation method. Besides, the stability analysis of the proposed method is analyzed and the accuracy of it is examined with $$L_{\infty }$$ and $$L_2$$ norm errors. At the end, the results obtained by the proposed method are compared with the methods given in previous works and it indicates an improvement in comparison with previous works.

Numerical solution of time-fractional fourth-order reaction-diffusion model arising in composite environments

The fractional reaction-diffusion equation has an important physical and theoretical meaning, but its analytical solution poses considerable problems. This paper develops an efficient numerical process, the local radial basis function generated by the finite difference (named LRBF-FD) method, for finding the approximation solution of the time-fractional fourth-order reaction-diffusion equation in the sense of the Riemann-Liouville derivative. The time fractional derivative is approximated using the second-order accurate formulation, while the spatial terms are discretized by means of the LRB-FFD method. The advantage of the local collocation method is to approximate the differential operators by a weighted sum of the values of the function on a local set of nodes (local support) by deriving the RBF expansion. Furthermore, the ill-conditioned matrix resulting from using the global collocation method is avoided. The unconditional stability property and convergence analysis of the time-discrete approach are thoroughly proven and verified numerically. Three numerical examples are presented confirming the theoretical formulation and effectiveness of the new approach.

Numerical analysis of the fractional evolution model for heat flow in materials with memory

This paper develops the solution of the two-dimensional time fractional evolution model using finite difference scheme derived from radial basis function (RBF-FD) method. In this discretization process, a finite difference formula is implemented to discrete the temporal variable, while the local RBF-FD formulation is utilized to approximate the spatial variable. The pattern of data distribution in the local support domain is assumed as having a fixed number of nodes. The local RBF-FD is based on the local support domain that leads to a sparsity system and also avoids the ill-conditioning problem caused by global collocation method. The stability and convergence of time-discrete approach in H1-norm are discussed by means of the energy method. Numerical results illustrate the proposed method and demonstrate that it provides accurate solutions on regular and irregular computational domains.

Numerical approximation of the time fractional cable model arising in neuronal dynamics

The cable equation is one useful description for modeling phenomena such as neuronal dynamics and electrophysiology. The time-fractional cable model (TFCM) generalizes the classical cable equation by considering the anomalous diffusion that occurs in the ionic motion present for example in the neuronal system. This paper proposes a novel meshless numerical procedure, the radial basis function-generated finite difference (RBF-FD), to approximate the TFCM involving two fractional temporal derivatives. The time discretization of the TFCM is performed based on the Grunwald–Letnikov expansion. The spatial derivatives are discretized using the RBF-FD. The pattern of data distribution in the support domain is assumed as having a fixed number of points. The RBF-FD is based on the local support domain that leads to a sparsity system and also tackles the ill-conditioning problem caused by global collocation method. The theoretical stability and the convergence analysis of the scheme are also discussed in detail. It is shown that the proposed method is efficient and that the numerical results confirm the theoretical formulation.

Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem

An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal.
 Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods.
 The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems.
 The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order of convergence, usual for any well designed spectral algorithm, to an algebraic one.
 As expected, the accuracy of ChC outcomes overpasses by far that of FEM outcomes.

A Radial Basis Function-Generated Finite Difference Method to Evaluate Real Estate Index Options

The local radial basis functions (RBF) method is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. The purpose of this paper is to design and describe the valuation of the real estate index options by a local RBF scheme based multiquadric radial basis function-generated finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and removes the difficulty of the ill-conditioned conventional global collocation methods. This paper employs an optimal variable shape parameter for the multiquadric basis functions at each grid point of the domain. Meanwhile, a local mesh refinement technique is adopted to deal with non-smooth payoffs of option. These techniques are effective and stable in improving the computational accuracy of the RBF-FD method. Several numerical experiments are presented and compared with the FD and compactly supported RBF methods to demonstrate the good performances of the proposed method. Lastly, the RBF-FD method is extended to price the American option of the real estate index.

A meshless local collocation method for time fractional diffusion wave equation

In this manuscript, we present a radial basis function based local collocation method for solving time fractional diffusion-wave equation. The advantage of the local collocation method is only the discretization points located in each sub-domain, surrounding the collocation point, need to be considered. It also avoids the ill-conditioning problem arises due to the use of global collocation method. The theoretical stability and convergence of the proposed time semi-discrete scheme are also discussed. Numerical experiments for one dimension and two dimension problems are carried out. It is shown that the present method is efficient and numerical results have nice agreements with theoretical result.

Sinc-collocation method for solving sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates

This paper seeks to develop an efficient method for solving natural convection of a non-Newtonian nanofluid flow between two vertical flat plates. Sodium alginate is considered as the non-Newtonian fluid, and then two distinct types of nanoparticles, namely silver and copper, are added to it. To do so, the governing boundary layer and temperature equations are reduced to a set of ordinary differential equations. This approach is based on a global collocation method using Sinc basis functions, and the resulting set of ordinary differential equations are replaced by a system of algebraic equations. It is well known that the Sinc procedure converges to the solution at an exponential rate. Numerical results are included to demonstrate the validity and applicability of the method, and a comparison is made with the existing results. Also, the effect of various parameters such as Prandtl number (Pr), dimensionless non-Newtonian viscosity number $$(\delta )$$ and nanoparticle volume fraction ( $$\phi $$ ) on non-dimensional velocity and temperature profiles are discussed. It was concluded from this study that velocity and temperature increased with increasing Pr. Moreover, our results indicate that both the velocity and temperature decrease as $$\delta $$ increases. Finally, the results demonstrated that, when $$\phi $$ increases, the velocity increases but the temperature values decrease.

A new method for evaluating options based on multiquadric RBF-FD method

In this paper, a new local meshless approach based on radial basis functions (RBFs) is presented to price the options under the Black–Scholes model. The global RBF approximations derived from the conventional global collocation method usually lead to ill-conditioned matrices. Employing the idea of local approximants of the finite difference (FD) method and combining it with the radial basis function (RBF) method can result in a local meshless approach such as RBF-FD. It removes the difficulty of ill-conditionness of the original method. The new proposed approach is unconditionally stable as it is shown by Von-Neumann stability analysis. It is fast and produces high accurate results as shown in numerical experiments. Moreover, we took into account the variation of shape parameter and analyzed numerically the behavior of the RBF-FD method.

CAD-based collocation eigenanalysis of 2-D elastic structures

This paper investigates the performance of the global collocation method for the numerical eigenfrequency extraction of 2-D elastic structures. The method is applied to CAD-based macroelements, starting from the older blending function Coons-Gordon interpolation (based on Lagrange polynomials) and extending to tensor product Bézier and B-splines. Numerical findings show equivalence between Lagrangian and Bézierian macroelements, while a mass lumping procedure is proposed for the former ones. Concerning B-splines, the influence of multiplicity of inner knots and the position of collocation points is thoroughly investigated. The theory is supported by 2-D numerical examples on rectangular and curvilinear structures of simple shape under plane stress conditions, in which the approximate solution rapidly converges towards the exact solution faster than that of the conventional finite element of similar mesh density.

Piecewise spectral collocation method for system of Volterra integral equations

The main purpose of this paper is to investigate the piecewise spectral collocation method for system of Volterra integral equations. The provided convergence analysis shows that the presented method performs better than global spectral collocation method and piecewise polynomial collocation method. Numerical experiments are carried out to confirm these theoretical results.

An nth-order shear deformation theory for static analysis of functionally graded sandwich plates

This paper focuses for the first time on the static analysis of sandwich plates with functionally graded faces and homogeneous core by an nth-order shear deformation theory and meshless global collocation method. The meshless global collocation method approximates the solutions of governing differential equations based on the nth-order shear deformation theory using all nodes in the problem domain. The deflection and stress of a simply supported sandwich plates under sinusoidal load are calculated to verify the accuracy and efficiency of the present theory.

A New Stable Local Radial Basis Function Approach for Option Pricing

In this paper, we develop a new local meshless approach based on radial basis functions (RBFs) to solve the Black---Scholes equation. The global RBF approximations derived from conventional global collocation method usually lead to ill-conditioned matrices. The new scheme employs the idea of the finite difference method to localize them. It removes the difficulty of ill-conditioning of the original method. The new proposed approach is unconditionally stable as it is shown by Von-Neumann stability analysis. As well as it is fast and it produces accurate results as shown in numerical experiments.

B-spline collocation method for boundary value problems in complex domains

In this paper, an over-determined, global collocation method based upon B-spline basis functions is presented for solving boundary value problems in complex domains. The method was truly meshless approach, hence simple and efficient to programme. In the method, any governing equations were discretised by global B-spline approximation as the B-spline interpolants. As the interpolating B-spline basis functions were chosen, the present method also posed the Kronecker delta property allowing boundary conditions to be incorporated efficiently. The present method showed high accuracy for elliptic partial differential equations in arbitrary domain with Neumann boundary conditions. For coupled Poisson problems with complex Neumann boundary conditions, the boundary collocation approach was adopted and applied in a simple and less costly manner to further improve the accuracy and stability. Applications from elasticity problems were given to demonstrate the efficacy and capability of the present method. In addition, the relation between accuracy and stability for the method was better justified by the new effective condition number given in literature.

Numerical Wave Solutions for Nonlinear Coupled Equations using Sinc-Collocation Method

In this paper, numerical solutions for nonlinear coupled Korteweg-de Vries(abbreviated as KdV) equations are calculated by the Sinc-collocation method. This approach is based on a global collocation method using Sinc basis functions. The first step is to discretize time derivative of the KdV equations by a classic finite difference formula, while the space derivatives are approximated by a weighted scheme. Sinc functions are used to solve these two equations. Soliton solutions are constructed to show the nature of the solution. The numerical results are shown to demonstrate the efficiency of the newly proposed method.

Static Analysis of Laminated Composite Plates Using n-Order Shear Deformation Theory and a Meshless Global Collocation Method

In this article, an n-order shear deformation theory is used to analyze the static characteristic of laminated composite plates. The third-order theory of Reddy can be considered as a special case of present n-order theory (n = 3). Governing equations and boundary conditions expressed in terms of strong form based on the present n-order theory are discretized by a meshless global collocation method. Maximum deflection and stress of the simply-supported laminated plate under sinusoidal load are compared with available published results, which demonstrate the accuracy and efficiency of present n-order theory.