Non-smooth Solutions Research Articles

Spectral Collocation Methods for Fractional Integro‐Differential Equations with Weakly Singular Kernels

In this paper, we propose and analyze a spectral approximation for the numerical solutions of fractional integro‐differential equations with weakly kernels. First, the original equations are transformed into an equivalent weakly singular Volterra integral equation, which possesses nonsmooth solutions. To eliminate the singularity of the solution, we introduce some suitable smoothing transformations, and then use Jacobi spectral collocation method to approximate the resulting equation. Later, the spectral accuracy of the proposed method is investigated in the infinity norm and weighted L2 norm. Finally, some numerical examples are considered to verify the obtained theoretical results.

Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

An inverse source problem for a non-automonous time fractional diffusion equation of order \((0<\beta <1)\) is considered in a bounded Lipschitz domain in \(\mathbb {R}^d\). The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe’s method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from \(t = 0\). Rothe’s method on a uniform grid addresses the existence of a such a solution (non-smooth with \(t^\gamma \) term where \(1>\gamma > \beta \)) under low regularity assumptions, whilst Rothe’s method over graded grids has the advantage to cope better with the behaviour at \(t=0\) (also here \(t^\beta \) is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

Stable numerical schemes for time-fractional diffusion equation with generalized memory kernel

The paper aims to develop the stable numerical schemes for generalized time-fractional diffusion equations (GTFDEs) with smooth and non-smooth solutions on the non-uniform grid. In time, the generalized Caputo derivative is discretized by a difference scheme of order (2−α) on a non-uniform grid where 0<α<1. Choosing the non-uniform meshes in the case of the smooth and non-smooth solution is also essential, so we graded the mesh in both cases separately. Stability and convergence for smooth as well as non-smooth solutions are obtained in L2-norm and L∞-norm respectively. Several numerical results are presented to show how the grading of meshes is essential. Also, numerical results validate the efficiency and effectiveness of proposed schemes and show how a non-uniform grid produces better results.

A numerical study of 21-cm signal suppression and noise increase in direction-dependent calibration of LOFAR data

ABSTRACT We investigate systematic effects in direction-dependent gain calibration in the context of the Low-Frequency Array (LOFAR) 21-cm Epoch of Reionization (EoR) experiment. The LOFAR EoR Key Science Project aims to detect the 21-cm signal of neutral hydrogen on interferometric baselines of 50–250 λ. We show that suppression of faint signals can effectively be avoided by calibrating these short baselines using only the longer baselines. However, this approach causes an excess variance on the short baselines due to small gain errors induced by overfitting during calibration. We apply a regularized expectation–maximization algorithm with consensus optimization (sagecal-co) to real data with simulated signals to show that overfitting can be largely mitigated by penalising spectrally non-smooth gain solutions during calibration. This reduces the excess power with about a factor of 4 in the simulations. Our results agree with earlier theoretical analysis of this bias-variance trade off and support the gain-calibration approach to the LOFAR 21-cm signal data.

Levenberg–Marquardt method with general convex penalty for nonlinear inverse problems

We consider a Levenberg–Marquardt method for solving nonlinear inverse problems in Hilbert spaces. The proposed method uses general convex penalty terms to reconstruct nonsmooth solutions of inverse problems. Instead of an a priori choice, the regularization parameter in each iteration is chosen by solving an equation which depends on the residual. We utilize the discrepancy principle to terminate the iteration and give the convergence results. In addition, numerical simulations are presented to test the performance of the method.

Fast L2-1[formula omitted] Galerkin FEMs for generalized nonlinear coupled Schrödinger equations with Caputo derivatives

The paper is concerned with the unconditional stability and optimal error estimates of Galerkin finite element methods (FEMs) for a class of generalized nonlinear coupled Schrödinger equations with Caputo-type derivatives. We improve the results in [1] to a higher order temporal scheme by using a type of new Grönwall inequality. By introducing a time-discrete system, the error is separated into two parts: the temporal error and the spatial error. As the result of τ-independent of the spatial error, we obtain the L∞-norm boundedness of the fully discrete solutions without any restrictions on the grid ratio. The unconditionally optimal L2-norm error estimate is then obtained naturally. Furthermore, in order to numerically solve the system with nonsmooth solutions, we construct another Galerkin FEM with nonuniform temporal meshes, and corresponding fast algorithm by using sum-of-exponentials technique is also built. Finally, numerical results are reported to show the accuracy and efficiency of the proposed FEMs and the corresponding fast algorithms.

A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions

An inverse source problem for non-smooth multiterm time Caputo fractional diffusion with fractional order designed as β0 < β1 < ⋯ < βM < 1 is the case of study in a bounded Lipschitz domain in $\mathbb {R}^{d}$ . The missing solely time-dependent source function is reconstructed from an additional integral measurement. The existence, uniqueness and regularity of a weak solution for the inverse source problem is investigated. We design a numerical algorithm based on Rothe’s method over graded meshes, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the multiterm time Caputo fractional subdiffusion problem is that the solution possibly lacks the smoothness near the initial time, although it would be smooth away from t = 0. In this contribution, we will establish an extension of Grönwall’s inequalities for multiterm fractional operators. This extension will be crucial for showing the existence of a unique solution to the inverse problem. The theoretical obtained results are supported by some numerical experiments.

A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

Uniqueness for Inverse Source Problems of Determining a Space-Dependent Source in Time-Fractional Equations with Non-Smooth Solutions

In this contribution, we investigate an inverse source problem for a fractional diffusion and wave equation with the Caputo fractional derivative of the space-dependent variable order. More specifically, we discuss the uniqueness of a solution when reconstructing a space-dependent source from a time-averaged measurement, or a final in time measurement. Weakly singular solutions are included in the class of admissible solutions. The obtained results are also valid if the order of the fractional derivative is constant.

On the Calculation of the Moore–Penrose and Drazin Inverses: Application to Fractional Calculus

This paper presents a third order iterative method for obtaining the Moore–Penrose and Drazin inverses with a computational cost of O(n3), where n∈N. The performance of the new approach is compared with other methods discussed in the literature. The results show that the algorithm is remarkably efficient and accurate. Furthermore, sufficient criteria in the fractional sense are presented, both for smooth and non-smooth solutions. The fractional elliptic Poisson and fractional sub-diffusion equations in the Caputo sense are considered as prototype examples. The results can be extended to other scientific areas involving numerical linear algebra.

Asymptotic linear - nonlinear duality, indeterminism and mathematical intelligence

The formalism of asymptotic duality structure with applications to late time asymptotic states of linear and nonlinear systems is presented. A scale invariant linear differential system is shown to admit higher derivative discontinuous, nonsmooth solutions under the action of asymptotic duality transformations. The relevant asymptotic quantities are self dual and finitely differentiable. The geometrical meaning and applications to higher order nonlinear systems are presented on the space of differentiable functions. A novel duality relationship between linear and nonlinear systems is exposed. A linear system is shown to extend self similarly over nonlinear, classically inaccessible scales thereby extending classical smooth solution space to a nonsmooth one. A nonlinear system, on the other hand, is shown to proliferate into a system of self similar linear equations that could be solved exactly to have a very efficient high precision computation of periodic orbits. An extension of self dual asymptotics to critically self dual case in the space of continuous functions reveals a connection to prime number theorem and the associated correction term respecting the Riemann hypothesis.

Spatial convergence study of Lax-Friedrichs WENO fast sweeping method on the SN transport equation with nonsmoothness

Spatial convergence study of the Lax-Friedrichs fast sweeping method based on the classical third order WENO scheme (WENO3) without a priori smoothness indicators is performed on the variants of Larsen’s benchmark problem with nonsmooth solutions. For comparison, third order upwind (UPWD3), weighted diamond difference, step method, bilinear discontinuous finite element method are also included. In optically thick regimes of the cases with continuous angular fluxes, WENO3 detects steep gradients of angular fluxes and handles them as discontinuities to suppress the unphysical oscillation, which improves the accuracy significantly compared with UPWD3. In the cases with discontinuous angular fluxes, WENO3 achieves non-oscillatory solutions with limited numerical diffusion. Nonlinear Gauss-Seidel iteration is necessary to obtain convergent solution for the nonlinearity of WENO3, which leads to low computational efficiency. The convergence behavior of nonlinear Gauss-Seidel iteration is affected by nonsmoothness of the exact solution, mesh number and the relaxation parameter.

Stability analysis and novel solutions to the generalized Degasperis Procesi equation: An application to plasma physics.

In this work two kinds of smooth (compactons or cnoidal waves and solitons) and nonsmooth (peakons) solutions to the general Degasperis-Procesi (gDP) equation and its family (Degasperis-Procesi (DP) equation, modified DP equation, Camassa-Holm (CH) equation, modified CH equation, Benjamin-Bona-Mahony (BBM) equation, etc.) are reported in detail using different techniques. The single and periodic peakons are investigated by studying the stability analysis of the gDP equation. The novel compacton solutions to the equations under consideration are derived in the form of Weierstrass elliptic function. Also, the periodicity of these solutions is obtained. The cnoidal wave solutions are obtained in the form of Jacobi elliptic functions. Moreover, both soliton and trigonometric solutions are covered as a special case for the cnoidal wave solutions. Finally, a new form for the peakon solution is derived in details. As an application to this study, the fluid basic equations of a collisionless unmagnetized non-Maxwellian plasma is reduced to the equation under consideration for studying several nonlinear structures in the plasma model.

Fourth-order compact difference schemes for the two-dimensional nonlinear fractional mobile/immobile transport models

In this article, we develop fourth-order compact difference schemes based on the linearized generalized BDF2-θ to solve the two-dimensional nonlinear fractional mobile/immobile (M/I) transport equations. We derive theoretical results, including unconditional stability and error estimates associated with solution regularity. Finally, we provide extensive numerical examples with smooth solutions to demonstrate the effectiveness and accuracy of the schemes and develop the corrected compact difference scheme to recovery convergence results with nonsmooth solutions.

Numerical solution of two-dimensional weakly singular Volterra integral equations with non-smooth solutions

Various numerical methods have been proposed for the solution of weakly singular Volterra integral equations but, for the most part, authors have dealt with linear or one-dimensional weakly singular Volterra integral equations, or have assumed that these equations have smooth solutions. The main purpose of this paper is to propose and analyse a numerical method for the solution of two-dimensional nonlinear weakly singular Volterra integral equations of the second kind. In general the solutions of these equations exhibit singularities in their derivatives at t=0 even if the forcing functions are smooth. To overcome these difficulties a simple smoothing change of variables is proposed. By applying this transformation an equation is obtained which, while still being weakly singular, can have a solution as smooth as is required. We then solve this transformed integral equation using Navot’s quadrature rule for computing integrals with an end point singularity. A new extension of a discrete Gronwall inequality allows us to prove convergence and obtain an error estimate. The theoretical results are then verified by numerical examples.

Numerical solutions for variable-order fractional Gross–Pitaevskii equation with two spectral collocation approaches

Abstract This paper addresses the numerical solution of multi-dimensional variable-order fractional Gross–Pitaevskii equations (VOF-GPEs) with initial and boundary conditions. A new scheme is proposed based on the fully shifted fractional Jacobi collocation method and adopting two independent approaches: (i) the discretization of the space variable and (ii) the discretization of the time variable. A complete theoretical formulation is presented and numerical examples are given to illustrate the performance and efficiency of the new algorithm. The superiority of the scheme to tackle VOF-GPEs is revealed, even when dealing with nonsmooth time solutions.

Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes

This article devotes to developing robust and simple correction techniques with efficient algorithms for a class of second-order time stepping methods, namely the shifted fractional trapezoidal rule (SFTR), for subdiffusion problems to resolve the initial singularity and nonlocality. The stability analysis and sharp error estimates in terms of the smoothness of the initial data and source term are presented. As a byproduct in numerical tests, we find amazingly that the Crank–Nicolson scheme ( $$\theta =\frac{1}{2}$$ ) without initial corrections can restore the optimal convergence rate for the subdiffusion problem with smooth initial data and source terms. To deal with the nonlocality, fast algorithms are considered to reduce the computational cost from $$O(N^2)$$ to $$O(N \log N)$$ and save the memory storage from O(N) to $$O(\log N)$$ , where N denotes the number of time levels. Numerical tests are performed to verify the sharpness of the theoretical results and confirm the efficiency and accuracy of initial corrections and the fast algorithms.

Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations

In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.

Use of very weak approximate boundary layer solutions to spatially nonsmooth singularly perturbed problems

We consider singularly perturbed Dirichlet problems which are, in the simplest nontrivial case, of the typeε2u″(x)=f(x,u(x)) for x∈[0,1],u(0)=u0,u(1)=u1. For small ε>0 we prove existence and local uniqueness of solutions u=uε, which are close to functions of the typeu¯ε(x)=u¯(x)+φ0(x/ε)+φ1((1−x)/ε) with f(x,u¯(x))=0 for x∈[0,1] and withφ0″(ξ)=f(0,u¯(0)+φ0(ξ)) for ξ∈[0,∞),u¯(0)+φ0(0)=u0,φ0(∞)=0,φ1″(ξ)=f(1,u¯(1)+φ1(ξ)) for ξ∈[0,∞),u¯(1)+φ1(0)=u1,φ1(∞)=0, and we show that ‖uε−u¯ε‖∞→0 for ε→0. We do not suppose monotonicity of the correctors φ0 and φ1. And, mainly, we do not suppose any regularity besides continuity of the functions f(⋅,u) and u¯. Hence, the functions u¯ε approximately satisfy the boundary value problem for ε≈0 in a very weak sense only, and one cannot expect that ‖uε−u¯ε‖∞=O(ε) for ε→0, in general. For the proofs we use an abstract result of implicit function theorem type which was designed for applications to spatially nonsmooth singularly perturbed boundary value problems with nonsmooth approximate solutions.