Integrated radial basis functions (IRBFs) to simulate nonlinear advection–diffusion equations with smooth and non-smooth initial data

The subdiffusion equation with a Caputo fractional derivative of order � 2 (0, 1) in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for dis- cretizing the Caputo fractional derivative in time. The scheme was analyzed earlier independently by Lin and Xu (2007) and Sun and Wu (2006), and an O(� 2−� ) convergence rate was established, under the assumption that the solution is twice continuously differentiable in time. However, in view of the smoothing property of the subdiffusion equation, this regularity condition is restrictive, since it does not hold even for the homogeneous problem with a smooth initial data. In this work, we revisit the error analysis of the scheme, and establish an O(�) convergence rate for both smooth and nonsmooth initial data. The analysis is valid for more gen- eral sectorial operators. In particular, the L1 scheme is applied to one-dimensional space-time fractional diffusion equations, which involves also a Riemann-Liouville derivative of order � 2 (3/2,2) in space, and error estimates are provided for the fully discrete scheme. Numerical experiments are provided to verify the sharpness of the error estimates, and robustness of the scheme with respect to data regularity.

Parallel radial basis function methods for the global optimization of expensive functions

Purpose – The purpose of this paper is to solve the interface discontinuities in radial basis function (RBF) method for multi-medium boundary value problems (BVPs). The discontinuity of the solution derivatives is not easily handled with RBF method because of infinitely smoothness. Design/methodology/approach – The essence of solving BVP is to construct the continuous potential function surfaces. Hence, from constructing surface aspect, this paper proposed and compared the global and subzone schemes for RBF method. Their implementation schemes and mathematic models can then be derived. Numerical experiments and comparison are carried out for electric and magnetic field calculation. Findings – In the numerical experiments, the subzone scheme has shown its significant advantageous, it can approximate not only the potential function but also its derivative on interface boundary with high accuracy. So the physical characteristics of discontinuities on the interface can be revealed clearly. The overall precision is significantly improved. Originality/value – This paper proposed an effective subzone scheme for RBF method in multi-medium BVP. It is an improvement for RBF method based on its domain decomposition idea. And it is also a candidate for solving complex BVP.

Integrated multiquadric radial basis function approximation methods

Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added. In this paper we show that it is possible to remove the Gibbs phenomenon from RBF approximations of discontinuous functions as well as from RBF solutions of some hyperbolic partial differential equations. Although the theory for the resolution of the Gibbs phenomenon by reprojection in Gegenbauer polynomials relies on the orthogonality of the basis functions, and the RBF basis is not orthogonal, we observe that the Gegenbauer polynomials recover high order convergence from the RBF approximations of discontinuous problems in a variety of numerical examples including the linear and nonlinear hyperbolic partial differential equations. Our numerical examples using multi-quadric RBFs suggest that the Gegenbauer polynomials are Gibbs complementary to the RBF multi-quadric basis.

An accurate and stable RBF method for solving partial differential equations

Placement

We study the space-time finite element discretizations for time fractional parabolic integro-differential equations in a bounded convex polygonal domain in $${\mathbb {R}}^d (d=1,2,3)$$ . Both spatially semidiscrete and fully discrete finite element approximations are considered and analyzed. We use piecewise linear and continuous finite elements to approximate the space variable whereas the time discretization uses two fully discrete schemes based on the convolution quadrature, namely the backward Euler and the second-order backward difference. For the spatially discrete scheme, optimal order a priori error estimates are derived for smooth initial data, i.e., when $$u_0\in H_0^1(\varOmega )\cap H^2(\varOmega )$$ . Moreover, for the homogeneous problem, almost optimal error estimates for positive time are established for nonsmooth initial data, i.e., when the initial function $$u_0$$ is only in $$ L^2(\varOmega )$$ . The error estimates for the fully discrete methods are shown to be optimal in time for both smooth and nonsmooth initial data under the specific choice of the kernel operator in the integral. Finally, we provide some numerical illustrations to verify our theoretical analysis.

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v ∈ L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

This research work presents a newly truly meshless approach based upon the integrated radial basis functions (IRBFs) technique to study the fractional modified anomalous sub‐diffusion equation. First, the temporal direction is discretized by a finite difference method with second‐order accuracy. The stability analysis and convergence of the proposed time‐discrete formulation are investigated, theoretically. Then, the spatial direction is approximated by the IRBFs methodology. In this approach, the largest‐order derivative is approximated by a linear combination of RBFs and then the lower‐order derivative and also the unknown function are constructed by repeated integration. According to this procedure, the existing derivatives in the main fractional PDE are approximated smoothly. Furthermore, to show the efficiency of the developed numerical procedure, we employed some non‐rectangular computational domains to obtain the numerical results. On the other hand, the mentioned model is solved in one‐ two‐, and three dimensional cases. The numerical results confirm the ability and efficiency of the new numerical method for solving time fractional PDEs on complex computational domains.

A stochastic computing approach is implemented in the present work to solve the nonlinear nanofluidics system that occurs in the model of atomic physics. The process converts the partial differential nanofluidics system with suitable level of similarities transformation into nonlinear systems of differential equations. For the construction of datasets, finite difference scheme (Lobatto IIIA) is applied through different selection of collocation points for nonlinear nanofluidics system having accuracy of order four. Lobatto IIIA has a strong point to tackle extremely nonlinear systems of ordinary differential equations in smooth way. For different scenarios, datasets are well trained through computing scheme to investigate the heat transfer and thermal performance of nanofluidic transportation system of nanofluids and hybrid nanofluids toward stretching surfaces with variation of Biot number, Nusselt number and skin fraction. Furthermore, the reliability, accuracy and efficiency are endorsed through various statistical analysis and graphical illustrations of proposed computing scheme.

A meshless RBF method for computing a numerical solution of unsteady Burgers’-type equations

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations.

Non-linear parameter estimation in pesticide degradation

An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations