Non-rectangular Domains Research Articles

A hybrid kernel-based meshless method for numerical approximation of multidimensional Fisher’s equation

We propose and analyze a meshless method of lines by considering some hybrid radial kernels. These hybrid kernels are constructed by linearly combining infinite smooth radial functions to piecewise smooth radial functions; which are then used for spatial approximation on trial spaces spanned by translates of positive definite radial functions. After spatial approximation, a high-order ODE solver is invoked for efficient and stable time-integration of the resultant semi-discrete system of ordinary differential equations (ODEs). Unlike the mesh-based method of lines, the proposed method works for arbitrary scattered data points and is equally effective for problems over non-rectangular domains. The proposed method is tested on one-, two- and three-dimensional reaction–diffusion Fisher equation for its numerical stability, accuracy, and efficiency against the contemporary meshless and mesh-based methods. The economical computational cost, improved accuracy, eigenvalues stability, and well-conditioning of system matrices are observed against RBF collocation and RBF-PS methods.

Data assimilation in 2D hyperbolic/parabolic systems using a stabilized explicit finite difference scheme run backward in time

ABSTRACTAn artificial example of a coupled system of three nonlinear partial differential equations generalizing 2D thermoelastic vibrations, is used to demonstrate the effectiveness, as well as the limitations, of a non iterative direct procedure in data assimilation. A stabilized explicit finite difference scheme, run backward in time, is used to find initial values, [u(.,0),v(.,0),w(.,0)], that can evolve into a useful approximation to a hypothetical target result [u∗(.,Tmax),v∗(.,Tmax),w∗(Tmax)], at some realistic 0 $ ]]>Tmax>0. Highly non smooth target data are considered, that may not correspond to actual solutions at time Tmax. Stabilization is achieved by applying a compensating smoothing operator at each time step. Such smoothing leads to a distortion away from the true solution, but that distortion is small enough to allow for useful results. Data assimilation is illustrated using 512×512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Computational experiments show that efficient FFT-synthesized smoothing operators, based on (−Δ)q with real q>3, can be successfully applied, even in nonlinear problems in non-rectangular domains. However, an example of failure illustrates the limitations of the method in problems where Tmax, and/or the nonlinearity, are not sufficiently small.

Innovative Method for Computing Approximate Solutions of Non-Homogeneous Wave Equations with Generalized Fractional Derivatives

In this work, a well-known non-homogeneous wave equation with temporal fractional derivative is approximately investigated. A recently defined generalized non-local fractional derivative is utilized as the fractional operator. A novel technique is proposed to approximate the solutions of wave equation with generalized fractional derivative. The proposed method is based on the shifted Chebyshev polynomials and a combination of collocation and residual function methods. Theoretical analysis of the convergence of the proposed method is performed. Approximate solutions are derived in both rectangular and non-rectangular (general) domains.

Analysis of plane elasticity problems using the dual mesh control domain method

The dual mesh control domain method (DMCDM) proposed by Reddy combines the advantages of the finite element method (interpolation of variables) and the finite volume method (satisfaction of the global form of the governing equations). Currently, the method has been used to solve two-dimensional heat transfer, fluid and solid mechanics problems on structured meshes of linear right triangular and bilinear rectangular elements. However, such meshes are not useful from the standpoint of most practical problems which are posed on irregular domains. The present study extends the DMCDM for the analysis of plane elasticity problems involving non-rectangular domains meshed with arbitrarily shaped elements. The approach makes use of the isoparametric formulation, in which the primal mesh used for the approximation of the geometry is also used for approximating the dependent unknowns. Several boundary value problems of elasticity are solved using the DMCDM, and the results are compared with those obtained using the finite element method (FEM). The results show that the DMCDM exhibits excellent convergence and accuracy while requiring less computational cost when compared with the FEM.

Consistent travelling waves solutions to the non-linear time fractional Klein–Gordon and Sine-Gordon equations through extended tanh-function approach

In this study, the extended tanh-function method has been used to find further general travelling wave solutions for space-time fractional nonlinear partial differential equations, namely, the time fractional nonlinear Sine-Gordon equation and Klein–Gordon equation. The mentioned equations are useful for explaining a series of experimental seismic data, modelling of strain waves, and structures correlated to fault dynamics and the subduction slab, as well as slow earthquakes, episodic tremor, slow slip events, and tremor migration patterns. This method is used to attain exact solutions to a variety of nonlinear partial differential equations that were spatially temporal fractional. To get analytical solutions of the travelling wave type of certain nonlinear evolution equations, a power series in tanh was first utilized as an ansatz. These nonlinear fractional partial differential equations (NLFPDEs) are solved on a variety of non-rectangular domains. By using the fractional complex transform and the properties of the confirmable derivative, the two equations are reduced to ordinary differential equations. The solutions are sketched in 3D, and contour patterns, including king type, single soliton, double solitons, multiple solitons, bell shape, and other sorts of solutions. The achieved solutions are very much effective for explaining the above-stated phenomena.

NUMERICAL SOLUTION OF TRAVELING WAVES IN CHEMICAL KINETICS: TIME-FRACTIONAL FISHERS EQUATIONS

This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis functions to compute space derivatives while Caputo derivative scheme utilizes for time-fractional integration to semi-discretize the model equations. To assess the accuracy, maximum error norm is used. In order to validate the proposed method, some non-rectangular domains are also considered.

The meshless local Petrov–Galerkin method based on moving Taylor polynomial approximation to investigate unsteady diffusion–convection problems of anisotropic functionally graded materials related to incompressible flow

This paper concerns to a meshless local Petrov–Galerkin (MLPG) method for studying the unsteady diffusion–convection problems of anisotropic functionally graded materials. A new version of MLPG method based on the moving Taylor polynomial approximation is developed to discrete the spatial variable. Then, we obtain a system of ODEs which depends to the time variable. A strong stability preserving (SSP) Runge–Kutta idea is provided to solve the final ODEs with enough accuracy and stability. Also, the grading function which defines the variable elastic coefficient can be any types of continuous functions. The developed numerical formulation is applied for different examples of non-rectangular domains to check its accuracy.

Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and meshfree methods

The nonlinear Sine-Gordon equation is one of the widely used partial differential equations that appears in various sciences and engineering. The main purpose of writing this article is providing an efficient numerical method for solving two-dimensional (2D) time-fractional stochastic Sine–Gordon equation on non-rectangular domains. In this method, radial basis functions (RBFs) and finite difference scheme are used to calculate the approximate solution of the mentioned problem. The complexity of solving this problem arises from its high dimension, irregular area, stochastic and fractional terms. Finite difference technique is applied to overcome on the problem dimension, whereas interpolation method based on RBFs is the best idea for solving problems defined in irregular domains. The stochastic Sine–Gordon equation is transformed into elliptic stochastic differential equations using the finite difference method and meshfree method based on RBFs are used to approximate the obtained stochastic differential equation. Some numerical examples are included to investigate the efficiency and accuracy of the presented method.

Computation of Asymptotic Spectral Distributions for Sequences of Grid Operators

The asymptotic spectral properties of matrices of grid operators on polygonal domains in the plane are studied in the case of refining triangular grids. Methods available for analyzing spectral distributions are largely based on tool of the theory of generalized locally Toeplitz sequences (GLT theory). In this paper, we show that the matrices of grid operators on nonrectangular domains do not form GLT sequences. A method for calculating spectral distributions in such cases is proposed. Generalizations of GLT sequences are introduced, and preconditioner based on them are proposed.

Solution of Maxwell’s Equations for Nonrectangular Electromagnetic Applications

The use of a fourth-order modified Runge–Kutta (MRK) scheme on a transformed coordinate system with Maxwell’s equations for nonrectangular domains and applications is presented. Maxwell’s equations are the governing equations for modeling electromagnetic wave propagation involving scattering, radiating structures, transmission lines, radar, biomedical applications, and nondestructive testing. Because of complex geometries in most problems where the material boundaries are not parallel to the grid axis, the application of finite differencing schemes in physical coordinates becomes nonviable. Therefore, by transforming the arbitrary-shaped structures to a uniform rectangular grid, numerical schemes and boundary conditions can be easily implemented. Numerical results for four cases are presented in this paper. Accuracy of the scheme has been established by comparing numerical results with the exact solution and error distribution plots. In the third case, scattering from the lossless and the lossy square cylindrical dielectric device where the plane wave source is injected using the total-field–scattered-field technique has been investigated. The results are compared with the results obtained from the benchmark finite difference time domain scheme. Lastly, the application of the numerical model to simulate scattering from curved boundaries has been presented in the fourth example.

Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences

More recently, it is discovered in the field of applied sciences and engineering that the telegraph equation is better suited to model reaction-diffusion than the ordinary diffusion equation. In this article, the second-order hyperbolic telegraph equations are analyzed numerically by means of an efficient local differential quadrature method utilizing the radial basis functions. The explicit time integration technique is used to semi-discretize the model in the time direction, while the space derivatives are discretized by the proposed meshless procedure. To test the accuracy and capabilities of the method, five test problems are considered utilizing both rectangular and non-rectangular domains, which show that the proposed scheme solutions are converging extremely quick in comparison with the different existing numerical techniques in the recent literature.

Direct meshless local Petrov-Galerkin method to investigate anisotropic potential and plane elastostatic equations of anisotropic functionally graded materials problems

The meshless local technique is employed to simulate potential problems involving material anisotropy and also plane elastostatic equations of anisotropic functionally graded materials (PEE-AFGM). We utilize direct meshless local Petrov Galerkin (DMLPG) technique to discrete the spatial directions. The DMLPG5 (based on weak form) is applied to solve potential problems involving material anisotropy and the DMLPG2 (based on strong form) is employed to simulate PEE-AFGM. The coefficients of the anisotropic conductivity require the gradient term to be modified accordingly. The grading function can be any general function that varies smoothly from point to point in the material. The developed procedures are tested on various examples on non-rectangular domains to check their accuracy.

Global in Time Results for a Parabolic Equation Solution in Non-rectangular Domains

This article deals with the parabolic equation ∂tw − c(t)∂2x w = f in D, D = { (t, x) ∈ R2 : t > 0, φ1 (t) < x < φ2(t) } with φi : [0,+∞[→ R, i = 1, 2 and c : [0,+∞[→ R satisfying some conditions and the problem is supplemented with boundary conditions of Dirichlet-Robin type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f ∈ L2(D) there exists a unique solution w such that w, ∂tw, ∂jw ∈ L2(D), j = 1, 2. Notice that the case of bounded non-rectangular domains is studied in [9]. The proof is based on energy estimates after transforming the problem in a strip region combined with some interpolation inequality. This work complements the results obtained in [19] in the case of Cauchy-Dirichlet boundary conditions

On the conservation of finite difference WENO schemes in non-rectangular domains using the inverse Lax-Wendroff boundary treatments

We discuss the issue of conservation of the total mass for finite difference WENO schemes solving hyperbolic conservation laws on a Cartesian mesh using the inverse Lax-Wendroff boundary treatments in arbitrary physical domains whose boundaries do not coincide with grid lines. The numerical fluxes near the boundary are suitably modified so that strict conservation of the total mass is achieved and the high order accuracy and non-oscillatory performance are not compromised. The key point is a suitable definition of the total mass, which is consistent with the high order accuracy finite difference framework over an arbitrary domain with a boundary not necessarily coinciding with grid lines. Extensive numerical examples are provided to demonstrate that our modified method is strictly conservative, and is high order accurate and has as good performance as the original high order WENO schemes with the Lax-Wendroff boundary treatments, for both smooth problems and problems with discontinuities, in both one- and two-dimensional problems involving both scalar equations and systems.

Practical uncertainty quantification analysis involving statistically dependent random variables

This article presents a practical refinement of generalized polynomial chaos expansion for uncertainty quantification under dependent input random variables. Unlike the Rodrigues-type formula, which exists for select probability measures, a three-step computational algorithm is put forward to generate a sequence of any approximate measure-consistent multivariate orthonormal polynomials. For uncertainty quantification analysis under dependent random variables, two regression methods, comprising existing standard least-squares and newly developed partitioned diffeomorphic modulation under observable response preserving homotopy (D-MORPH), are proposed to estimate the coefficients of generalized polynomial chaos expansion for the very first time. In contrast to the existing regression devoted so far to the classical polynomial chaos expansion, no tensor-product structure is required or enforced. The partitioned D-MORPH regression is applicable to either an underdetermined or overdetermined system, thus substantially enhancing the ability of the original D-MORPH regression. Numerical results obtained for Gaussian and non-Gaussian probability measures with rectangular or non-rectangular domains point to highly accurate orthonormal polynomials produced by the three-step algorithm. More significantly, the generalized polynomial chaos approximations of mathematical functions and stochastic responses from solid-mechanics problems, in tandem with the partitioned D-MORPH regression, provide excellent estimates of the second-moment properties and reliability from only hundreds of function evaluations or finite element analyses.

Local meshless differential quadrature collocation method for time-fractional PDEs

This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of $ \alpha \in (0,1) $ and $ \alpha\in(1,2) $. To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.

On the numerical solution of nonlinear integral equations on non-rectangular domains utilizing thin plate spline collocation method

This article investigates an approximate scheme to solve nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The integral equations considered in the current paper are considered together with either smooth or weakly singular kernels. The offered method utilizes thin plate splines as a basis in the discrete collocation method. We can regard thin plate splines as a type of the free shape parameter radial basis functions. These basis functions establish an accurate and stable technique to estimate an unknown function by using a set of scattered points on the solution domains. Since the thin plate splines have limited smoothness, the integrals appeared in the scheme cannot be estimated by classical integration rules. Therefore, we introduce a special precise quadrature formula on non-rectangular domains to compute these integrals. The proposed scheme does not require any mesh generations, so it is meshless and does not depend on the domain form. Error analysis is also provided for the method. The performance and convergence of the new approach are tested on four two-dimensional integral equations given on the wing, mushroom, pentagon and fish-like domains.

Analysis of mixed finite element method (MFEM) for solving the generalized fractional reaction–diffusion equation on nonrectangular domains

In the current manuscript, we consider a generalized fractional reaction–diffusion equation. The considered model is based on the time fractional derivative. The developed scheme is based on two procedures. At first, we obtain a semi-discrete scheme for the temporal direction. The time fractional derivative is discretized by using a difference scheme with second-order accuracy. Thus, the final time-discrete formula has second-order accuracy in the temporal direction. Then, we obtain a fully discrete scheme by applying the mixed finite element method (MFEM). For the constructed numerical technique, we prove the unconditional stability and also obtain an error bound for the full-discrete scheme by using the energy method. We employ some test problems to show the accuracy of the proposed technique.

Solving two-dimensional integral equations of the second kind on non-rectangular domains with error estimate

In this paper, the collocation method is applied on two-dimensional integral equations of the second kind on non-rectangular domains. Since the domains of these equations are non-rectangular and so directly applying the collocation method for them is difficult, at first, the integral equations are converted to equivalent integral equations on rectangular domains. Then, two-dimensional Jacobi collocation method is applied. Furthermore, an error estimate for the method is investigated and several examples demonstrate the accuracy and efficiency of the method.

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

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.