Sobolev Equations Research Articles

A hybrid high-order method for Sobolev equation with convection-dominated term

We consider a new hybrid high-order method for the Sobolev equation with convection-dominated term. We adopt the Pk polynomials to approximate the primary unknown on mesh elements and faces where k is a given positive integer. The stability and high accuracy are obtained by using reconstruction to formulate the gradient term and convective term and by adding a suitable stabilization term. We analyze the semi-discrete formulation of the hybrid high-order method and use Crank-Nicolson difference in time to get the corresponding fully-discrete scheme. We prove stability and convergence of the approximate solution in energy norm. The energy errors hold irrespective of the reciprocal of dispersion, diffusion and reaction coefficients. Numerical results underline the efficiency of the proposed methods and our analytical findings.

Inverse Source Problem for Sobolev Equation with Fractional Laplacian

In this paper, we are interested in the problem of determining the source function for the Sobolev equation with fractional Laplacian. This problem is ill-posed in the sense of Hadamard. In order to edit the instability instability of the solution, we applied the fractional Landweber method. In the theoretical analysis results, we show the error estimate between the exact solution and the regularized solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Finally, we investigate the convergence of the source function when fractional order β ⟶ 1 + .

Linearly compact scheme for 2D Sobolev equation with Burgers’ type nonlinearity

Linearly compact scheme for 2D Sobolev equation with Burgers’ type nonlinearity

The reduced-dimension technique for the unknown solution coefficient vectors in the Crank–Nicolson finite element method for the Sobolev equation

In this paper, we mainly deal with the reduced-dimension of coefficient vectors of unknown solutions for the Crank–Nicholson finite element (CNFE) method of the Sobolev equation. For this purpose, we first establish the CNFE methods of the Sobolev equation in both functional form and matrix form, and provide the existence, stability, and error estimates of CNFE solutions. We then build a CNFE reduced-dimension recursive (CNFERDR) method in the matrix-form by using a proper orthogonal decomposition (POD) technique, and discuss the existence, stability, and error estimates of CNFERDR solutions by using matrix analysis. In this case, the CNFERDR method includes only few unknowns, but has the same basis functions and accuracy as the CNFE method. Finally, we use two numerical examples to verify the effectiveness of the CNFERDR method.

A hybrid high-order method for the Sobolev equation

In this paper, we present a new hybrid high-order method for the Sobolev equation. For a given positive integer k, we adopt the Pk polynomials to approximate the discrete unknown solutions on mesh elements and faces. In our method, we use reconstruction to formulate the gradient term and add suitable stabilization term to enforce the stability and high order accuracy. We analyze the full-discrete formulation of the hybrid high-order method. The stability, energy error estimate with k+1 order and L2 estimate with k+2 order are proved. At last, numerical results show the efficiency of our method.

A comparative study on interior penalty discontinuous Galerkin and enriched Galerkin methods for time-fractional Sobolev equation

A comparative study on interior penalty discontinuous Galerkin and enriched Galerkin methods for time-fractional Sobolev equation

Numerical investigation of the variable-order fractional Sobolev equation with non-singular Mittag–Leffler kernel by finite difference and local discontinuous Galerkin methods

In this paper to approximate the Heydari–Hosseininia non-singular fractional derivative, we construct the L1-2 discretization by providing the error estimate. The error estimation of the L1 formula is also presented. The scheme uses the local discontinuous Galerkin method combing with the L1/L1-2 formula as spatial and time discretizations, respectively. To investigate the efficiency and accuracy of our scheme, variable-order fractional ordinary differential and 2-dimensional Sobolev equations are proposed. The scheme is second/third-order accurate in time for the L1/L1-2 formula, respectively. Utilizing k, the approximation degree, the rates of convergence in space are reported k+1 when time step chosen τ=hk+12 and τ=hk+13. Our argument is that new approximation L1-2 has less computational cost than the L1 discretization and numerical results would be given to confirm this reduction.

A hybrid approach established upon the Müntz‐Legender functions and 2D Müntz‐Legender wavelets for fractional Sobolev equation

This article proposes a hybrid technique for finding approximation solutions of the fractional 2D Sobolev equation. In the proposed approach, the Müntz‐Legender functions and Müntz‐Legender wavelets are, respectively, utilized to approximate the solution of the problem under consideration in the time and spatial directions. By implementing the presented technique, solving the 2D fractional Sobolev equation is converted into solving a system of algebraic equations. Three examples are solved to examine the validity of the proposed method.

Numerical simulation of 3-D Sobolev equation via local meshless method

In this study, we use an effective meshless method to estimate the numerical solution of 3-D time-fractional Sobolev equation. The recommended meshless method is used for the spatial derivatives while the Liouville-Caputo derivative technique is utilized for the time derivative portion of the model equation. Accuracy of the method is assessed via error norms and comparison is made with the exact solution and other numerical methods given in more current literature, which demonstrated that the suggested strategy produces excellent performance and is more computationally efficient.

One-parameter orthogonal spline collocation methods for nonlinear two-dimensional Sobolev equations with time-variable delay

This paper deals with the numerical solutions of initial–boundary value problems (IBVPs) of nonlinear two-dimensional Sobolev equations with time-variable delay. For solving this type of IBVPs, two classes of one-parameter orthogonal spline collocation (OSC) methods are constructed. For the obtained one-parameter OSC methods, their errors in L2- and H1-norm are analyzed and thus the corresponding error estimates are derived. With some numerical experiments, the computational effectiveness and theoretical accuracy of the methods are further confirmed. Moreover, a numerical comparison shows that the both kinds of one-parameter OSC methods have almost the same computational efficiency and accuracy under the same parameter and stepsizes.

Completely discrete schemes for 2D Sobolev equations with Burgers’ type nonlinearity

Completely discrete schemes for 2D Sobolev equations with Burgers’ type nonlinearity

Proper orthogonal decomposition Pascal polynomial-based method for solving Sobolev equation

PurposeThis study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover, this method does not require establishing grids in the computational domain.Design/methodology/approachIn this study, the authors present a meshfree method based on Pascal polynomial expansion for the numerical solution of the Sobolev equation. In general, Sobolev-type equations have several applications in physics and mechanical engineering.FindingsThe authors use the Crank-Nicolson scheme to discrete the time variable and the Pascal polynomial-based (PPB) method for discretizing the spatial variables. But it is clear that increasing the value of the final time or number of time steps, will bear a lot of costs during numerical simulations. An important purpose of this paper is to reduce the execution time for applying the PPB method. To reach this aim, the proper orthogonal decomposition technique has been combined with the PPB method.Originality/valueThe developed procedure is tested on various examples of one-dimensional, two-dimensional and three-dimensional versions of the governed equation on the rectangular and irregular domains to check its accuracy and validity.

Mountain pass solution to a perturbated Hardy–Sobolev equation involving p-Laplacian on compact Riemannian manifolds

ABSTRACT In this paper, we consider the following quasilinear equation: where M is a compact Riemannian manifold with dimension without boundary, and . Here , and are continuous functions on M satisfying some further conditions. The operator is the p-Laplace–Beltrami operator on M associated with the metric g, and is the Riemannian distance on . Moreover, we assume , , and with . The notion is the critical Hardy–Sobolev exponent. With the help of Mountain Pass Theorem, we get the existence results under different assumptions.

Time decay rates for the generalized MHD-α equations in Sobolev–Gevrey spaces

This paper proves the existence of a unique global in time solution for the generalized Magnetohydrodynamics-α equations in Sobolev–Gevrey (and Sobolev) spaces (including critical cases) if it is assumed that the initial data is small enough. In addition, the authors show that this same solution is analytic and study its long time behavior.

An accurate approach based on the orthonormal shifted discrete Legendre polynomials for variable-order fractional Sobolev equation

This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family of basis functions, are employed to generate an operational matrix method for this equation. A new fractional operational matrix related to these polynomials is extracted and employed to construct the presented method. Using this approach, an algebraic system of equations is obtained instead of the original variable-order equation. The numerical solution of this system can be found easily. Some numerical examples are provided for verifying the accuracy of the generated approach.

Superconvergence error estimate of Galerkin method for Sobolev equation with Burgers' type nonlinearity

In this paper, based on the implicit Euler scheme in the temporal direction, the superconvergence property is investigated by using the special property of the bilinear element on the rectangular mesh for the Sobolev equation with Burgers' nonlinearity. The existence and uniqueness of the fully-discrete solution is proved. Further, the superconvergence error estimate in L∞(H1)-norm is established in terms of a novel approach, i.e., the technique of the combination of the interpolation operator and projection operator. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

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.

A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics

This paper develops a numerical approach for finding the approximate solution of the Sobolev model. This model describes many natural processes, such as thermal conduction for different media and fluid evolution in soils and rocks. The proposed method approximates the unknown solution with the help of two main stages. At a first stage, the time discretization is performed by means of a second-order finite difference procedure. At a second stage, the space discretization is accomplished using the local radial basis function partition of unity collocation method based on the finite difference (LRBF-PUM-FD). The major disadvantage of global techniques is the high computational burden of solving large linear systems. The LRBF-PUM-FD significantly sparsifies the linear system and reduces the computational burden, while simultaneously maintaining a high accuracy level. The time-discrete formulation is studied in terms of the stability and convergence analysis via the energy method. Three examples are illustrated to verify the efficiency and accuracy of the method.

The study of a continuous Galerkin method for Sobolev equation with space-time variable coefficients

In this article, we will apply a space-time continuous Galerkin (CG) method to solve the numerical solution of Sobolev equation with space-time variable coefficients. Both the spatial and temporal variables of this method are discretized by finite element (FE) method, and hence it can easily attain the high order precision both in space and time directions as well as have the good stability. In addition, the variable spatial grid structures from one time span to the next and the time steps are permitted for this method, which are more suitable to design adaptive algorithm on unstructured mesh. We detailedly give the well-posed analysis of the numerical solution and the a priori error estimation in L∞(H1) norm without any restrictions on the space-time grid ratio. Finally, some numerical experiments are showed to conform the convergence orders and validate the high efficiency for the method investigated here.