Nonlinear Sobolev Equations Research Articles

Efficient spectral and spectral element methods for Sobolev equation with diagonalization technique

In this paper, we first introduce a new series of Legendre basis functions by using the matrix decomposition technique, which are simultaneously orthogonal in both L2- and H1-inner products. Then, we construct efficient space-time spectral method for linear Sobolev equation using spectral approximation in space and multi-domain collocation approximation in time, which can be implemented in a synchronous parallel fashion. Next, we propose a novel Legendre spectral element method for solving nonlinear Sobolev equation with Burgers' type term, which reduce the non-zero entries of linear systems and computational cost. Some rigorous error estimates are carried out for one-dimensional Sobolev equation. Numerical experiments illustrate the effectiveness and accuracy of the suggested approaches.

On nonlinear Sobolev equation with the Caputo fractional operator and exponential nonlinearity

The initial value problem for the Caputo type time‐fractional Sobolev equation with a nonlinear exponential source function is investigated in this work. We establish the existence and uniqueness of mild solutions corresponding to two different initial data assumptions. We derive global results of a unique mild solution with small initial data using some Sobolev/Sobolev‐Orlicz embeddings, a weighted Banach space, and the fixed point theorem. In the absence of any smallness assumptions, the Cauchy iteration method demonstrates that the mild solution blows up at a finite time or exists globally in time. Finally, we consider some illustrated examples to test the results obtained in theory.

Legendre-finite difference method for solving fractional nonlinear Sobolev equation with Caputo derivative

The Sobolev equation is known as a partial differential equation with second-order partial derivatives. Initially, the Sobolev equation was used to solve the problem of turbulence in physics. However, due to its extensive use in mathematics and geometry, it soon became one of the most important mathematical equations. Despite the significance of this equation, its solution in fractional and nonlinear cases has received less attention in research. This paper presents a numerical algorithm for solving fractional nonlinear Sobolev model using Legendre polynomials and a finite difference scheme. Firstly, the finite difference plan is employed to discretize the time variable, where stability and convergence of the method are demonstrated. The stability of the method is dependent on the nonlinear part of the problem. The two-dimensional Legendre polynomials are utilized at collocation points to discretize the spatial variables. Finally, several illustrative examples are conducted to evaluate the accuracy of the proposed approach.

Numerical analysis of a space-time continuous Galerkin method for the nonlinear Sobolev equation

In this work, we shall use space-time continuous Galerkin (STCG) method to analyze the nonlinear Sobolev equation. This method is different from the conventional finite element methods, that is, it not only uses finite element to discrete spatial variables but also uses finite element to discrete temporal variable. Therefore it is easier to obtain high order accuracy in time and space than the conventional finite element methods and has the good stability. Also, it does not need time step size and space mesh parameter to satisfy the stability conditions. In this paper, we show a detailed analysis including the proof of the existence and uniqueness of the numerical solution and the a priori error estimate. Especially, the analysis method given here can be similarly applied to the more general nonlinear problems. At last, the numerical examples reveal that the iterative method is more efficient than the linearized method for the nonlinear problems with STCG method. Also, the numerical experiments show that the STCG method is more efficient than the space-time discontinuous Galerkin (STDG) method in the practical calculations.

Virtual element method for nonlinear Sobolev equation on polygonal meshes

Virtual element method for nonlinear Sobolev equation on polygonal meshes

Two linearized second-order block-centered finite difference methods for nonlinear Sobolev equations

Two linearized second-order block-centered finite difference methods for nonlinear Sobolev equations

On Cauchy Problems for Nonlinear Sobolev Equations in Ferroelectricity Theory

Two Cauchy problems for the nonlinear Sobolev equations@ and @ are investigated. Conditions are found under which the Cauchy problems have weak generalized local-in-time solutions, and the blow-up conditions for weak solutions of these problems are determined.

On Cauchy Problems for Nonlinear Sobolev Equations in Ferroelectricity Theory

On Cauchy Problems for Nonlinear Sobolev Equations in Ferroelectricity Theory

Linearized compact difference methods for solving nonlinear Sobolev equations with distributed delay

AbstractThis paper deals with the numerical computation and analysis for nonlinear Sobolev equations with distributed delay. We present linearized compact difference methods for solving one‐ and two‐dimensional cases, respectively. The solvability, convergence and stability of the methods are derived and it is shown under some suitable conditions that the methods have second‐order accuracy in time and fourth‐order accuracy in space. Moreover, with several numerical examples, the theoretical accuracy and computational effectiveness of the proposed methods are confirmed.

Localized meshless approaches based on theta method and BDF2 for nonlinear Sobolev equation arising from fluid dynamics

This paper formulates the second-order BDF and theta schemes based on an efficient localized meshless technique for computing the approximate solution of the Sobolev model. The model is one of the useful description for modeling the flow of fluids through fractured rocks, movement of moisture in soils. This proposed approach discretizes the nonlinear Sobolev equation in two main steps. In the first step, temporal discretization is accomplished with the help of the second-order BDF and theta schemes, respectively. In the second step, the localized radial basis function partition of unity collocation method (L-RBF-PUCM) is used for spatial discretization, thereby constructing a fully-discrete algorithm. For large linear problems, global techniques have the disadvantage of high computational cost. The proposed method overcomes this shortcoming well and reduces the computational burden by sparsifying the linear system. The convergence analysis and unconditional stability of the semi-discrete formulation are examined by means of the energy method. Numerical results are carried out to verify the validity and accuracy of the L-RBF-PUCM.

Maximum error estimates of two linearized compact difference schemes for two-dimensional nonlinear Sobolev equations

In this paper, two classes of high-order numerical schemes on the time discretization for the solutions of two-dimensional nonlinear Sobolev equations are analyzed. The two-level Newton linearized difference scheme had been established in the previous literature, in which the error estimates in L2-norm and H1-norm are obtained, but that in L∞-norm remains unsolved. The second difference scheme is established based on a three-level linearized difference technique in temporal direction. The numerical discretization in spatial dimension for both schemes utilizes the compact difference operator. We prove that the orders of convergence for both difference schemes in L∞-norm are O(τ2+h14+h24) based on the energy argument, where τ denotes the temporal step size, h1 and h2 denote the spatial step sizes in x-direction and y-direction, respectively. The numerical tests further verify the theoretical results and demonstrate the efficiency of both difference schemes.

A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation

In this paper, one class of finite difference scheme is proposed to solve nonlinear space fractional Sobolev equation based on the Crank-Nicolson (CN) method. Firstly, a fractional centered finite difference method in space and the CN method in time are utilized to discretize the original equation. Next, the existence, uniqueness, stability, and convergence of the numerical method are analyzed at length, and the convergence orders are proved to be O τ 2 + h 2 in the sense of l 2 -norm, H α / 2 -norm, and l ∞ -norm. Finally, the extensive numerical examples are carried out to verify our theoretical results and show the effectiveness of our algorithm in simulating spatial fractional Sobolev equation.

Approximate solution of two-dimensional Sobolev equation using a mixed Lucas and Fibonacci polynomials

A numerical scheme based on polynomials and finite difference method is developed for numerical solutions of two-dimensional linear and nonlinear Sobolev equations. In this approach, finite difference method is applied for the discretization of time derivative whereas space derivatives are approximated by two-dimensional Lucas polynomials. Applying the procedure and utilizing finite Fibonacci sequence, differentiation matrices are derived. With the help of this technique, the differential equations have been transformed to system of algebraic equations, the solution of which compute unknown coefficients in Lucas polynomials. Substituting the unknowns constants in Lucas series, required solution of targeted equation has been obtained. Performance of the method is verified by studying some test problems and computing E2, E $$_{\infty }$$ and Erms (root mean square) error norms. The obtained accuracy confirms feasibility of the proposed technique.

Exact Solutions of the Equation of a Nonlinear Conductor Model

We consider a nonlinear Sobolev equation of the third order describing the electric field potential in a conductor. Six classes of exact solutions are constructed, and their qualitative behavior is analyzed.

On the analysis of a kind of nonlinear Sobolev equation through locally applied pseudo‐spectral meshfree radial point interpolation

AbstractIn this study, we develop an approximate formulation for two‐dimensional nonlinear Sobolev problems by focusing on pseudospectral meshless radial point interpolation (PSMRPI) which is a kind of locally applied radial basis function interpolation and truthfully a meshless approach. In the PSMRPI method, the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. The convergence and stability of the technique in some sense are studied via some examples to show the validity and trustworthiness of the PSMRPI technique.

A two-grid method for discontinuous Galerkin approximations to nonlinear Sobolev equations

A two-grid algorithm for discontinuous Galerkin approximations to nonlinear Sobolev equations is proposed. H1 norm error estimate of the two-grid method for the nonlinear parabolic problem is derived. The analysis shows that our two-grid discontinuous Galerkin algorithm will achieve asymptotically optimal approximation as long as the mesh sizes satisfy $h = O(H^{\frac {r+1}{r}})$ , where r is the order of the discontinuous finite element space. The numerical experiments are presented to prove the efficiency of our algorithm.

Exact Solutions to One Nonlinear Sobolev Equation

Exact Solutions to One Nonlinear Sobolev Equation

A block-centered finite difference method for the nonlinear Sobolev equation on nonuniform rectangular grids

In this article, a block-centered finite difference method for the nonlinear Sobolev equation is introduced and analyzed. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with superconvergence O(Δt+h2+k2) for scalar unknown p, its gradient u and its flux q are established on nonuniform rectangular grids, where Δt, h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H2‐norm by the temporal error. Superconvergence results of order O(h2 + τ3) in H1‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

Fully discrete approximation of general nonlinear Sobolev equations

We consider abstract quasilinear evolution equations of Sobolev type in a Hilbert setting. We propose two fully discrete schemes and prove some error estimates under minimal assumptions. Various examples that enter into our abstract framework are considered, for each of them our theoretical results are confirmed by several numerical experiments.