Non-smooth Solutions Research Articles

Direct Operational Vector Scheme for First-Kind Nonlinear Volterra Integral Equations and Its Convergence Analysis

In the current paper, an efficient direct numerical scheme is developed to approximate the solution of nonlinear Volterra integral equations of the first kind. This computational method is based upon operational matrices and vectors which eventually lead to the sparsity of the coefficients matrix of the obtained system. For this purpose, the operational vector for hybrid block pulse functions and Chebyshev polynomials is constructed. Hybrid functions are powerful tools to approximate functions locally, and capable to adjust the orders of block-pulse functions and Chebyshev polynomials to achieve highly accurate numerical solutions. Its simple structure to implement, low computational cost and perfect approximate solutions are the major points of the presented method. The convergence analysis of the numerical approach under the $$L^2$$ -norm is completely studied. Finally, numerical experiments indicate that the proposed approach is effective and powerful to deal with smooth and non-smooth solutions and verify the obtained theoretical results.

Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model

In this article, a time two-mesh (TT-M) algorithm combined with the H1-Galerkin mixed finite element (FE) method is introduced to numerically solve the nonlinear distributed-order sub-diffusion model, which is faster than the H1-Galerkin mixed FE method. The Crank-Nicolson scheme with TT-M algorithm is used to discretize the temporal direction at time $t_{n+\frac {1}{2}}$ , the FBN-𝜃 formula is developed to approximate the distributed-order derivative, and the H1-Galerkin mixed FE method is used to approximate the spatial direction. TT-M mixed element algorithm mainly covers three steps: first, the mixed finite element solution of the nonlinear coupled system on the time coarse mesh ΔtC is calculated; next, based on the numerical solution obtained in the first step, the numerical solution of the nonlinear coupled system on time fine mesh ΔtF is obtained by using Lagrange’s interpolation formula; finally, the numerical solution of the linearized system on time fine mesh ΔtF is solved by using the results in the second step. The existence and uniqueness of the solution for our numerical scheme are shown. Moreover, the stability and a priori error estimate are analyzed in detail. Furthermore, numerical examples with smooth and nonsmooth solutions are given to validate our method.

High Order Finite Difference Multi-resolution WENO Method for Nonlinear Degenerate Parabolic Equations

In this paper, we propose a new finite difference weighted essentially non-oscillatory (WENO) scheme for nonlinear degenerate parabolic equations which may contain non-smooth solutions. An alternative formulation is designed to approximate the second derivatives in a conservative form. In this formulation, the odd order derivatives at half points are used to construct the numerical flux, instead of the usual practice of reconstruction. Moreover, the multi-resolution WENO scheme is designed to circumvent the negative ideal weights and mapped nonlinear weights that appear when applying the standard WENO idea. We will describe the scheme formulation and present numerical tests for one- and two-dimensional, demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.

Numerical Methods for Integral Equations of the Second Kind With Non-Smooth Solutions of Bounded Variation

Numerical Methods for Integral Equations of the Second Kind With Non-Smooth Solutions of Bounded Variation

Unconditional convergence of linearized orthogonal spline collocation algorithm for semilinear subdiffusion equation with nonsmooth solution

AbstractA linearized orthogonal spline collocation (OSC) method withC1splines of degree≥3on a suitable graded mesh is formulated and analyzed for approximate solution of an initial‐boundary‐value problem of semilinear subdiffusion equations with nonsmooth solutions in time. The sharp error estimate in theL2norm is established without any restriction on the relative temporal and spatial mesh sizes. Such unconditional convergence results are proved by including the typical singularity of the solution near the timet = 0. Results of numerical experiments support the analytical results.

Generalized mapped nodal Laguerre spectral collocation method for Volterra delay integro-differential equations with noncompact kernels

This paper is devoted to solve a class of weakly singular Volterra integro-differential equations with noncompact kernels. Since the solution of such these models may has singularities, so the classical spectral methods lose their high accuracy for solving them. Innovation of this article is that, we use the generalized mapped nodal Laguerre spectral collocation method to deal with the singularity. Therefore, we can make good use of the advantages of the mapped Laguerre functions. The best advantage of the proposed method is its robustness for solving problems that have singularity near the left (or right) boundary of the computational interval. We present the construction and analysis of the generalized log orthogonal Laguerre functions collocation method in this paper and some numerical examples with nonsmooth solutions are included to show the efficiency of the suggested numerical scheme with respect to the classical Jacobi spectral methods.

Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey

Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for N=2, 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.

Adaptive Space-Time Finite Element Methods for Non-autonomous Parabolic Problems with Distributional Sources

Abstract We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable coefficients that are possibly discontinuous in space and time. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We present new a priori and a posteriori error estimates for low-regularity solutions. In order to avoid reduced rates of convergence that appear when performing uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators and functional a posteriori error estimators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by space-time algebraic multigrid. In particular, in the 4d space-time case, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.

Smoothing transformation and collocation methods for third-kind linear Volterra integral equations

In 2016, Sonia et al. first considered the convergence order for the third-kind linear Volterra integral equations (VIEs) based on the assumption that solutions are smooth. For the third-kind linear VIEs with nonsmooth solutions, we construct high-order numerical algorithms and discuss the convergence order. By introducing a new suitable independent variable, we obtain a transformed equation with a smooth exact solution. Then the solvability of the transformed equation is investigated on the basis of piecewise polynomial collocation methods. Meanwhile, the convergence order of the collocation solution is given. Furthermore, based on the inverse transformation, we get the convergence order of the original equation. Numerical simulations are finally presented to demonstrate the effectiveness of the theoretical results.

A spectral collocation method for nonlinear fractional initial value problems with nonsmooth solutions

A general class of nonlinear fractional differential equations is considered. Some sufficient conditions for the existence and uniqueness of exact solution are established by using Weissinger's fixed point theorem and the Gronwall‐Bellman lemma. A spectral collocation method based on the smoothing technique is presented to solve the problem numerically. Then the rigorous error estimates under the L2 and L∞ norms are derived. The most remarkable feature of the method is its capability to achieve spectral convergence for weakly singular solutions. Finally, numerical results are given to support the theoretical conclusions with smooth and weakly singular solutions.

A linearized high-order Galerkin finite element approach for two-dimensional nonlinear time fractional Klein-Gordon equations

In this paper, we propose a linearized finite element method for solving two-dimensional fractional Klein-Gordon equations with a cubic nonlinear term. The employed time discretization is a weighted combination of the L2 − 1σ formula introduced recently by Lyu and Vong (Numer. Algorithms 78(2):485–511, 2018), Galerkin finite element method is used for the spatial discretization, and the cubic nonlinear term is handled explicitly. Using mathematical induction, we prove that the numerical solution is bounded and the fully discrete scheme is convergent with second-order accuracy in time. In numerical experiments, some problems with both smooth and non-smooth exact solutions are considered.

Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations

We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.

Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes

In this paper, an unconditionally stable fully discrete numerical scheme for the two-dimensional (2D) time fractional variable coefficient diffusion equations with non-smooth solutions is constructed and analyzed. The L2-1σ scheme is applied for the discretization of time fractional derivative on graded meshes and anisotropic finite element method (FEM) is employed for the spatial discretization. The unconditional stability and convergence of the proposed scheme are proved rigorously. It is shown that the order O(h2+N−min{rα,2}) can be achieved, where h is the spatial step, N is the number of partition in temporal direction, r is the temporal meshes grading parameter and α is the order of fractional derivative. A numerical example is provided to verify the sharpness of our error analysis.

Incomplete iterative solution of subdiffusion

In this work, we develop an efficient incomplete iterative scheme for the numerical solution of the subdiffusion model involving a Caputo derivative of order $$\alpha \in (0,1)$$ in time. It is based on piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time and solves one linear algebraic system inexactly by an iterative algorithm at each time step. We present theoretical results for both smooth and nonsmooth solutions, using novel weighted estimates of the time-stepping scheme. The analysis indicates that with the number of iterations at each time level chosen properly, the error estimates are nearly identical with that for the exact linear solver, and the theoretical findings provide guidelines on the choice. Illustrative numerical results are presented to complement the theoretical analysis.

Finite Element Methods Based on Two Families of Second-Order Numerical Formulas for the Fractional Cable Model with Smooth Solutions

We apply two families of novel fractional $$\theta $$ -methods, the FBT- $$\theta $$ and FBN- $$\theta $$ methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional $$\theta $$ -methods, and the space direction is approximated by the finite element method. Some positivity properties of the coefficients for both of these methods are derived, which are crucial for the proof of the stability estimates. We analyse the stability of the scheme and derive an optimal convergence result with $$O(\tau ^2+h^{r+1})$$ for smooth solutions, where $$\tau $$ is the time mesh size and h is the spatial mesh size. Some numerical experiments with smooth and nonsmooth solutions are conducted to confirm our theoretical analysis. To overcome the singularity at initial value, the starting part is added to restore the second-order convergence rate in time.

Shifted Jacobi spectral-Galerkin method for solving fractional order initial value problems

In this paper, the shifted Jacobi spectral-Galerkin method is introduced to deal with fractional order initial value problems (FIVPs). In the proposed method, the exact solution of the FIVP is approximated by using shifted Jacobi polynomials on each subinterval of the total time. The main advantage of the proposed method is that the rate of convergence in L2-norm depends on the local smoothness of solution. This enables the proposed method to work for nonsmooth solutions. The second merit is that the flexibility between the length of sub-interval and the degree of polynomial significantly enhances the numerical accuracy. We derive the spectral-Galerkin approximation formula for the Volterra integral equation of the second kind which is equivalent to the FIVP. Error analysis in L2-norm for the case α=β=0 in shifted Jacobi polynomials is provided and it justifies the spectral rate of convergence with respect to both the degree and the length of the subinterval when the source function is Lipschitz continuous in the second argument. Numerical illustrations for multi-order, linear, and nonlinear FIVPs are demonstrated to confirm the convergence rate.

Approximation methods for the distributed order calculus using the convolution quadrature

In this article we generalize the convolution quadrature (CQ) method, which aims at approximating the fractional calculus, to the case for the distributed order calculus. Our method is a natural expansion that the approximation formulas, convergence results and correction technique reduce to the cases for the CQ method if the weight function \begin{document}$ \mu(\alpha) $\end{document} is defined by \begin{document}$ \delta(\alpha-\alpha_0) $\end{document} . Further, we explore a new structure of the solution of an ODE with the distributed order fractional derivative, which differs from those of the solutions of traditional fractional ODEs, and propose a new correction technique for this new structure to restore the optimal convergence rate. Numerical tests with smooth and nonsmooth solutions confirm our theoretical results and the efficiency of our correction technique.

Global consistency analysis of L1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations

Recently there has been a growing interest in designing efficient numerical methods for the solution of fractional differential equations. The solutions of such equations in general exhibit a weak singularity near the initial time. In this paper, we propose finite difference/spectral methods to solve the coupled nonlinear time-space fractional Schrödinger equations with non-smooth solutions in the time direction. The proposed methods combine the strength of the L1 scheme on both uniform and non-uniform grids and the Galerkin-Legendre method. The well-posedness of the numerical solution is proved. The convergence analysis shows clearly how the grading of the mesh and the regularity of the solution affect the order of convergence of the L1 scheme, so one can choose an optimal mesh grading which can be seen numerically by considering some numerical experiments.

A compact exponential difference method for multi-term time-fractional convection-reaction-diffusion problems with non-smooth solutions

This paper is concerned with a numerical method for a class of one-dimensional multi-term time-fractional convection-reaction-diffusion problems, where the differential equation contains a sum of the Caputo time-fractional derivatives of different orders between 0 and 1. In general the solutions of such problems typically exhibit a weak singularity at the initial time. A compact exponential finite difference method, using the well-known L1 formula for each time-fractional derivative and a fourth-order compact exponential difference approximation for the spatial discretization, is proposed on a mesh that is generally nonuniform in time and uniform in space. Taking into account the initial weak singularity of the solution, the stability and convergence of the method is proved and the optimal error estimate in the discrete L2-norm is obtained by developing a discrete energy analysis technique which enables us to overcome the difficulties caused by the nonsymmetric discretization matrices. The error estimate shows that the method has the spatial fourth-order convergence, and reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. The extension of the method to two-dimensional problems is also discussed. Numerical results confirm the theoretical convergence result, and show the applicability of the method to convection dominated problems.

An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions

This paper aims to provide a rigorous analysis of exponential convergence of an adaptive spectral collocation method for a general nonlinear system of rational-order fractional initial value problems. The key idea of the proposed method is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions to a class of fractional initial value problems, leading to spectrally accurate approximation. Numerical examples are presented, which verify the theoretical predictions and demonstrate that the new formulation of the spectral method leads to better performance compared to other known numerical approaches with a relatively smaller number of degree of freedoms.