Weak Singularity Research Articles

Solving Multi-Point Boundary Value Differential Equations with a Novel Semi-Analytical Algorithm

To address the general multi-point boundary value problems, we present a novel semi-analytical algorithm. This scheme is applicable to both linear and nonlinear fractional-order and integer-order differential equations under multi-point boundary value and/or initial value conditions. The proposed algorithm involves the introduction of auxiliary fractional equations derived from the main equation. Subsequently, a linear combination of the solutions to the auxiliary equations is formed to meet the boundary conditions of the main equation. The combination includes some unknown coefficients. In respect with these unknown coefficients, we optimize the relevant residual function to achieve its minimum value. This is done by using the method of weighted residuals. The convergence of the method is established and proved. The algorithm is employed to obtain approximate solutions for multi-point boundary and initial value problems. Additionally, it effectively addresses problems with solutions that exhibit weak singularities at the origin. Numerical results, including solving two bridge models, reveal that the proposed method demonstrates a substantial improvement in performance over existing methods.

A graded mesh technique for numerical approximation of a multi-term Caputo time-fractional Fokker-Planck equation in 2D space

This paper focuses on the development of an efficient numerical approach for solving a two-dimensional multi-term Caputo time fractional Fokker-Planck (TFFP) model. The solution of such problem, in general, shows a weak singularity at the time origin. A numerical technique based on a graded time mesh is proposed to handle the singular behaviour of the solution. The multi-term Caputo time fractional derivatives in the TFFP model are discretized by means of the L1 scheme on the nonuniform mesh, while a high-order compact alternating direction implicit finite difference scheme is designed to approximate the spatial derivatives. Convergence and stability analysis of the suggested method is analyzed. Two numerical examples subjected to smooth and nonsmooth solutions are presented to corroborate the theoretical results and illustrate the applicability of the method. The results obtained by the proposed graded mesh technique are compared with the results obtained by the uniform mesh technique.

Algorithms for Solving the Equilibrium Composition Model of Arc Plasma.

In the present study, the Homotopy Levenberg-Marquardt Algorithm (HLMA) and the Parameter Variation Levenberg-Marquardt Algorithm (PV-LMA), both developed in the context of high-temperature composition, are proposed to address the equilibrium composition model of plasma under the condition of local thermodynamic and chemical equilibrium. This model is essentially a nonlinear system of weakly singular Jacobian matrices. The model was formulated on the basis of the Saha and Guldberg-Waage equations, integrated with Dalton's law of partial pressures, stoichiometric equilibrium, and the law of conservation of charge, resulting in a nonlinear system of equations with a weakly singular Jacobian matrix. This weak singularity primarily arises due to significant discrepancies in the coefficients between the Saha equation and the Guldberg-Waage equation, attributed to differing chemical reaction energies. By contrast, the coefficients in the equations derived from the other three principles within the equilibrium composition model are predominantly single-digit constants, further contributing to the system's weak singularity. The key to finding the numerical solution to nonlinear equations is to set reasonable initial values for the iterative solution process. Subsequently, the principle and process of the HLMA and PV-LMA algorithms are analyzed, alongside an analysis of the unique characteristics of plasma equilibrium composition at high temperatures. Finally, a solving method for an arc plasma equilibrium composition model based on high temperature composition is obtained. The results show that both HLMA and PV-LMA can solve the plasma equilibrium composition model. The fundamental principle underlying the homotopy calculation of the (n-1) -th iteration, which provides a reliable initial value for the n-th LM iteration, is particularly well suited for the solution of nonlinear equations. A comparison of the computational efficiency of HLMA and PV-LMA reveals that the latter exhibits superior performance. Both HLMA and PV-LMA demonstrate high computational accuracy, as evidenced by the fact that the variance of the system of equations ||F|| < 1 × 10-15. This finding serves to substantiate the accuracy and feasibility of the method proposed in this paper.

Two linearized difference schemes on graded meshes for the time-space fractional nonlinear diffusion-wave equation with an initial singularity

Abstract In this article, two linearized difference schemes are considered to solve a class of time-space fractional nonlinear diffusion-wave equations with weak singularity near the initial time based on its equivalent integro-differential equation. The Crank-Nicolson method combined with the trapezoidal product integration rule with graded meshes is implemented to approximate the Riemann-Liouville integral. The fractional central difference formula and a novel compact difference method are used to construct fully discrete difference schemes. Using the energy method, the stability and convergence of the proposed schemes are demonstrated in terms of the L 2 norm with the accuracy of O N − min { r σ , 2 } + M − 2 and O N − min { r σ , 2 } + M − 4 , respectively. The theoretical analysis is complemented by numerical results.

Viscous Marangoni migration of an inviscid bubble by surfactant spreading: an exactly solvable model

A model is formulated of a two-dimensional migrating, or swimming, inviscid bubble in a viscous fluid whose unsteady displacement is caused by the spreading over its surface of an initial distribution of insoluble surfactant. Assuming small capillary and Reynolds numbers, and a linear equation of state giving the surface tension as a function of surfactant concentration, the quasi-steady Stokes flow around the bubble is found analytically and explicit formulas are determined for the time-dependent bubble speed and its final overall displacement. At infinite surface Péclet number this is done using a complex version of the method of characteristics to solve a complex partial differential equation of Burgers type. For a finite non-zero surface Péclet number, the problem is shown to be linearizable by a complex variant of the classical Cole–Hopf transformation. The formulation allows general statements to be made on the bubble speed and its total net displacement in terms of the initial surfactant distribution. A weak finite-time singularity in the surface activity associated with an isolated clean point on the bubble surface is also identified and studied in detail.

Local error analysis of L1 scheme for time-fractional diffusion equation on a star-shaped pipe network

Abstract The numerical analysis for differential equations on networks has become a significant issue in theory and diverse fields of applications. Nevertheless, solving time-fractional diffusion problem on metric graphs has been less studied so far, as one of the major challenging tasks of this problem is the weak singularity of solution at initial moment. In order to overcome this difficulty, a new L1-finite difference method considering the weak singular solution at initial time is proposed in this paper. Specifically, we utilize this method on temporal graded meshes and spacial uniform meshes, which has a new treatment at the junction node of metric graph by employing Taylor expansion method, Neumann-Kirchhoff and continuity conditions. Over the whole star graph, the optimal error estimate of this fully discrete scheme at each time step is given. Also, the convergence analysis for a discrete scheme that preserves the Neumann-Kirchhoff condition at each time level is demonstrated. Finally, numerical results show the effectiveness of proposed full-discrete scheme, which can be applied to star graphs and even more general graphs with multiple cross points.

A second-order fitted scheme for time fractional telegraph equations involving weak singularity

A second-order fitted scheme for time fractional telegraph equations involving weak singularity

Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh–Stokes equations

In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh–Stokes equations. The proposed two-grid FEM uses the L2-1σ scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The L2-norm and H1-norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy H=h12 and H=hr2r+2 respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2-1σ scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at t=0 is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with 0<α<1) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in L2 norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is O(τtsα−1+τ2α+h12+h22). Numerical experiments are given to verify the theoretical analysis.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

Higher order numerical approximations for non-linear time-fractional reaction–diffusion equations exhibiting weak initial singularity

In the present study, we introduce a high-order non-polynomial spline method designed for non-linear time-fractional reaction–diffusion equations with an initial singularity. The method utilizes the L2-1σ scheme on a graded mesh to approximate the Caputo fractional derivative and employs a parametric quintic spline for discretizing the spatial variable. Our approach successfully tackles the impact of the singularity. The obtained non-linear system of equations is solved using an iterative algorithm. We provide the solvability of the novel non-polynomial scheme and prove its stability utilizing the discrete energy method. Moreover, the convergence of the proposed scheme has been established using the discrete energy method in the L2 norm. It is proven that the method is convergent of order min{θα,2} in the temporal direction and 4.5 in the spatial direction, where α∈(0,1) denotes the order of the fractional derivative and the parameter θ is utilized in the construction of the graded mesh. Finally, we conduct numerical experiments to validate our theoretical findings and to illustrate how the mesh grading influences the convergence order when dealing with a non-smooth solution to the problem.

A Nyström method based on product integration for weakly singular Volterra integral equations with variable exponent

Weakly singular Volterra integral equations with variable exponent have integral kernels of the form (t−s)−α(t) with 0≤α(t)<1. A Nyström method based on product integration and interpolation by piecewise polynomials is constructed and analysed for the numerical solution of this class of problems. To deal with the weak singularity of typical solutions of such problems, suitably graded meshes are used. A rigorous error analysis proves convergence of the computed solution; moreover, superconvergence is obtained if the quadrature points are well chosen. These results also imply error bounds for the solution of a related iterated collocation method. Numerical experiments demonstrate the sharpness of our theoretical results for the Nyström method.

On second-order tensor representation of derivatives in shape optimization.

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class [Formula: see text]. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity

Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity

An efficient computational approach and its analysis for the Caputo time‐fractional convection reaction–diffusion equation in two‐dimensional space

This work concerns with a novel computational approach to solve a two‐dimensional Caputo time‐fractional convection reaction–diffusion (CTFCRD) equation with the weak singularity at the initial time. In this method, the ‐scheme on graded mesh is used to approximate the Caputo temporal derivative, while the spatial derivatives are approximated by using a compact finite difference method (CFDM) coupled with alternating direction implicit (ADI) scheme. We present a methodology to examine the optimal error estimates of the proposed scheme, in terms of ‐norm. Convergence and stability results are proved. It is shown that the suggested method yields an optimal order, that is, min(2‐ , , 2 +1) in time direction, where is the order of time fractional derivative and is the grading parameter and has a fourth‐order of convergence in space direction. Finally, two numerical examples are provided to verify the theoretical estimates and to illustrate the accuracy and efficiency of the method.

Nonlocal Changing-Sign Perturbation Tempered Fractional Sub-Diffusion Model with Weak Singularity

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables.

Failure prediction of an open-hole laminate under compression

Failure prediction of open-hole laminates under compression still remains a great challenge. A number of unique mechanics theories for composites developed by the author are successfully applied to analyze in plane compression induced various failures of the notched laminates in this paper. A buckling mode must be incorporated into the analysis of the compression induced delamination, and the rules for selecting a scale factor for the buckling analysis through ABAQUS are established. To simulate delamination, the interlaminar matrix stress modification method is applied. A more pertinent criterion for delamination is proposed. When and where is delamination initiated, how can the initiated delamination be propagated and how much is the delaminated area to be attained can be easily reproduced. Intralaminar failures are estimated based on Bridging Model and the matrix true stress theory. The ultimate strength of a notched laminate is assumed when any primary layer element outside neighborhoods of the stress singularity and weak singularity points firstly attains an ultimate failure. A limited, if not the minimum, number of inputs are required all measurable independently and following existing standards with no data calibration. Except for the pre-buckling analysis, no iteration is needed for prediction of all the other failures. The predicted failure modes and ultimate compressive loads of several laminates with single or double holes agree well with our measured counterparts.

Positive periodic solution of second-order difference equation with a repulsive singularity

A second-order difference equation with periodic boundary value conditions is studied in this article, the nonlinear term may be singular. A few conditions are given to guarantee the existence of positive periodic solution by fixed-point theorem in cones. Our results can be applied to both superlinearity and semilinearity in the case that the difference equation has no singularity, and can be applied to both strong singularity and weak singularity in the case that the difference equation has a repulsive singularity.

Discontinuous Galerkin finite element method for dynamic viscoelasticity models of power‐law type

AbstractLinear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior penalty Galerkin method (SIPG) for spatial discretization, and the implicit finite difference schemes in time, Crank–Nicolson method. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Grönwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well‐suited for long‐time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.

About a new nonlinear integro-differential equation of Volterra-Strum-Liouville type with a weakly singular kernel

This present paper proposes an analytical and numerical study of a new integro-differential Volterra-Strum-Liouville equation of the second type, having a weak singularity. We provide conditions that guarantee the existence and uniqueness of the solution of the nonlinear problem. Then, we develop a numerical technique using the production integration method. The numerical application shows the efficiency of the proposed procedure.