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This article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness. [ABSTRACT FROM AUTHOR]

The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms. [ABSTRACT FROM AUTHOR]

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone. [ABSTRACT FROM AUTHOR]

An alternative approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method (FEM) based on the concept of the Galerkin method is used. To construct finite-dimensional subspaces separate approximations of displacements, deformations, stresses, and their gradients are implemented by choosing the different sets of piecewise polynomial basis functions, interrelated by the stability condition of the mixed FEM approximation. This significantly simplifies the pre-requirement for approximating functions to belong to class C1 and allows one to use the simplest triangular finite elements with a linear approximation of displacements under uniform or near-uniform triangulation conditions. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses and their gradients are treated as local unknowns. The conditions of existence, uniqueness, and continuous dependence of the solution on the problem's initial data are formulated for discrete equations of mixed FEM. These are solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. This avoids the need to form a global stiffness matrix for the problem of strain gradient elasticity since it is enough to calculate only the residual vector in the current approximation. A set of modeling plane crack problems is solved. The obtained solutions agree with the results available in the relevant literature. Good convergence is achieved by refining the mesh for all scale parameters. All three problems under study exhibit specific qualitative features characterizing strain gradient solutions namely crack stiffness increase with length scale parameter and cusp-like closure effect. [ABSTRACT FROM AUTHOR]

The study aims to explore wavelet applications for analyzing nonlinear boundary value problems. Although several wavelet methods are reviewed in the literature, a comparative study of their strengths and limitations has found only a few attempts. This study bridges the gap between two wavelet-based numerical methods, namely, higher order Daubechies waveletbased Galerkin method and Haar wavelet collocation method, by conducting a comparative study. Nonlinear boundary value problems arising in mathematical physics are solved using both schemes, followed by the computation of optimal error estimates. Furthermore, the advantages offered by the Haar wavelet collocation method over the wavelet-Galerkin method and the rate of convergence are also discussed in detail. [ABSTRACT FROM AUTHOR]

• Detailed modeling of the Saint-Venant torsion problem for heterogeneous and orthotropic cross-sections. • Rigorous analy sis of well-posedness, regularity and Galerkin convergence of solutions for different continuous variational setting. • Proof, at a discrete level, that solutions satisfying the regularized problem also satisfy the discrete problem obtained by fixing a datum at a domain node. • Discussion about possible numerical issues when using the FEM solution technique of fixing a datum. • Show cases of different torsion problems available in the literature. For many years, torsion of arbitrary cross-sections has been a subject of numerous investigations from theoretical and numerical points of view. As it is well known, the resulting boundary value problem (BVP) governing such phenomenon happens to be a pure Neumann BVP and, therefore, its solutions are determined up to a constant. Among a large plethora of finite element method (FEM) techniques that can be used in this context, most of FEM practitioners resolve this uniqueness issue by fixing the candidate solution to a node of the domain. Although such popular and pinpointing technique is widely spread and works well for practical purposes, it does not have a continuous counterpart and therefore its justification remains a matter of debate. Hence, this self-contained work aims to address the modeling of arbitrary heterogeneous and orthotropic cross-sections as well as the theoretical and numerical aspects of their solutions. In particular, we discuss the existence of weak solutions, well-posedness, regularity of solutions, and convergence of Galerkin's method for different variational settings (with special focus on a regularized variational approach). Moreover, we establish a connection, at a discrete level, between the convergence of solutions of well-posed variational settings and those solutions coming from the usual practice of fixing a datum at a node. Finally, we discuss some numerical aspects of all the FEM discrete formulations proposed here by performing convergence analysis in L 2 and H 1 norms. The section of numerical results is closed by presenting a series of study cases ranging from a square cross-section composed of two different materials to an isotropic bridge cross-section for which no analytical solution exists. [ABSTRACT FROM AUTHOR]

In this article, we develop a meshfree numerical solver for fractional‐order governing differential equations. More specifically, we develop a mesh‐free interpolation‐based element‐free Galerkin numerical model for the fractional‐order governing differential equations. The proposed fractional element‐free Garlekin (f‐EFG) numerical model is a lighter and more accurate alternative to existing mesh‐based finite element solvers for the fractional‐order governing differential equations. We demonstrate here that the f‐EFG with moving least squares (MLS) interpolants are naturally suitable for the approximation of fractional‐order derivatives in terms of the corresponding nodal values, thereby alleviating several issues with FE solvers for such integro‐differential governing equations. We demonstrate the efficacy of the proposed numerical model for numerical solutions with benchmark problems on the linear and nonlinear elastic response of nonlocal elastic solid modeled via fractional‐order governing differential equations. However, it must be noted that the proposed f‐EFG algorithm can be extended to fractional‐order governing differential equations in diverse applications, including multiscale and multiphysics studies. [ABSTRACT FROM AUTHOR]

This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. By reducing the geometric differential equation to a system of algebraic equations, we achieve computational efficiency. The method not only proves to be computationally efficient but also showcases a valuable symmetric property, aligning with inherent patterns in geometric structures. This symmetry enhances the method's compatibility with the often-present symmetries in geometric boundary value problems, offering both computational advantages and a deeper understanding of geometric calculus. To demonstrate the reliability and efficiency of the proposed method, we present several examples with both homogeneous and non-homogeneous boundary conditions. These examples serve to validate the method's performance in practice. [ABSTRACT FROM AUTHOR]

This study investigates the vibration characteristics of rectangular cross-sectioned and straight beams with imperfect supports, focusing on the role of dry friction at the contact interfaces. The contact interactions are reduced to resultant point loads, and the friction at the contact interfaces is modelled using the Jenkins friction model, introducing nonlinearity into the system. These nonlinear terms are included in solution-dependent boundary conditions for the governing differential equation of beam vibration. Two cases are considered in detail and solved: one where the beam is tightened between rigid clamps at both ends and excited from the middle with a harmonic displacement function, and another where only one end is clamped with the other end free but excited with an imposed harmonic displacement. The governing differential equation is solved analytically, separating the motion into two distinct regimes - full-stick and full-slip, using the Galerkin method. The results acquired from this analytical model are then compared to those from a numerical model, which is built and solved using the finite element method combined with a frequency sweep and time-marching. • This study considers beams subjected to friction at the supports instead of idealised motion restrictions. • Effects of contact friction on Euler-Bernoulli beams with imperfect clamps are investigated analytically. • Friction is modelled as a displacement-dependent nonlinear force. • This analytical method can be used as an efficient tool for the prediction of the regime for systems with friction. [ABSTRACT FROM AUTHOR]

Over the last two decades, meshfree Galerkin methods have become increasingly popular in solid and fluid mechanics applications. A variety of these methods have been developed, each incorporating unique meshfree approximation schemes to enhance their performance. In this study, we examine the application of the Moving Least Squares and Local Maximum-Entropy (LME) approximations within the framework of Optimal Transportation Meshfree for solving Galerkin boundary-value problems. We focus on how the choice of basis order and the non-negativity, as well as the weak Kronecker-delta properties of shape functions, influence the performance of numerical solutions. Through comparative numerical experiments, we evaluate the efficiency, accuracy, and capabilities of these two approximation schemes. The decision to use one method over the other often hinges on factors like computational efficiency and resource management, underscoring the importance of carefully considering the specific attributes of the data and the intrinsic nature of the problem being addressed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem of the generalized pseudo‐parabolic equation containing viscoelastic terms and associated with Robin conditions. We establish first the local existence of solutions by the standard Galerkin method. Then, we prove blow‐up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow‐up rate for the weak solution by finding the upper bound and the lower bound for the blow‐up times and the upper bound and the lower bound for the blow‐up rate. For negative energy, we introduce a new method to prove blow‐up results with a sharper estimate for the upper bound for the blow‐up times. Finally, we prove both the global existence of the solution and the general decay of the energy functions under some restrictions on the initial data. [ABSTRACT FROM AUTHOR]

The reproducing kernel gradient smoothing integration (RKGSI) is an efficient technique to tackle the integration problem and optimal convergence in meshless methods. In this paper, the effect of the RKGSI on the element-free Galerkin (EFG) method is studied for elliptic boundary value problems with mixed boundary conditions. Theoretical results of smoothed gradients in the RKGSI are provided. Fundamental criteria on how to determine integration points and weights of quadrature rules are established according to necessary algebraic precision. By using the Nitsche's technique to impose Dirichlet boundary condition, the existence, uniqueness and error estimations of the solution of the EFG method with numerical integration are analyzed. Numerical results validate the theoretical analysis and the optimal convergence of the method. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative approach to solve a system of boundary value problems (BVPs), using the newly developed discontinuous Galerkin (DG) method, which eliminates the need for auxiliary variables. This work is the first in a series of papers on DG methods applied to partial differential equations (PDEs). By consecutively applying the DG method to each space variable of the PDE using the method of lines, we transform the problem into a system of ordinary differential equations (ODEs). We investigate the convergence criteria of the DG method on systems of ODEs and generalize the error analysis to PDEs. Our analysis demonstrates that the DG error's leading term is determined by a combination of specific Jacobi polynomials in each element. Thus, we prove that DG solutions are superconvergent at the roots of these polynomials, with an order of convergence of O (h p + 2) . [ABSTRACT FROM AUTHOR]

We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach. [ABSTRACT FROM AUTHOR]

The paper discusses how to verify the quality of approximate solutions to partial differential equations constructed by deep neural networks. A posterior error estimates of the functional type, that have been developed for a wide range of boundary value problems, are used to solve this problem. It is shown, that they allow one to construct guaranteed two-sided estimates of global errors and get distribution of local errors over the domain. Results of numerical experiments are presented for elliptic boundary value problems. They show that the estimates provide much more reliable information on the quality of approximate solutions generated by networks than the loss function, which is used as a quality criterion in the Deep Galerkin method. [ABSTRACT FROM AUTHOR]

We consider a space–time fractional parabolic problem. Combining a sinc quadrature-based method for discretizing the Riesz–Dunford integral with |$hp$| -FEM in space yields an exponentially convergent scheme for the initial boundary value problem with homogeneous right-hand side. For the inhomogeneous problem, an |$hp$| -quadrature scheme is implemented. We rigorously prove exponential convergence with focus on small times |$t$|⁠ , proving robustness with respect to startup singularities due to data incompatibilities. [ABSTRACT FROM AUTHOR]

In this work, we study the existence and uniqueness of the solution to a wave equation with dynamic Wentztell-type boundary conditions on a part of the boundary Γ1 of the domain Ω with non-linear delays in non-linear dampings in Ω and on Γ1, using the Faedo-Galerkin method. [ABSTRACT FROM AUTHOR]

The purpose of this study is to solve a nonlinear integral equation describing the propagation of electromagnetic waves in a body located in free space. The boundary-value problem for the Helmholtz equation is reduced to the solution of the integral equation. An iterative method of creating a nonlinear medium inside a body with a dielectric structure is constructed. The problem is solved numerically. The size of the matrix obtained in the calculation exceeds 30 000 elements. The internal convergence of the iterative method is shown. The plots illustrating the field distribution in the nonlinear body are presented. The numerical method has been proposed and implemented. [ABSTRACT FROM AUTHOR]

In this paper, we have developed a cubic B-spline technique to obtain numerical solutions for second-order linear singularly perturbed boundary value problems of convectiondiffusion type equations with discontinuous source terms. The solution to such types of problems contains boundaries and an interior layer. The error analysis of the proposed method has been studied and also proved the convergence of the method. The cubic B-spline technique has been implemented on two numerical examples which shows the efficiency and accuracy of the method. Obtained numerical results of the proposed method have compared with other existing methods and found that it gives better numerical solutions at the same number of mesh points. [ABSTRACT FROM AUTHOR]

In this paper, we solve a linear system of second-order boundary value problems by using the quadratic B-spline finite element method (FEM). The performance of the method is tested on one model problem. Comparisons are made with both the analytical solution and some recent results.The obtained numerical results show that the method is efficient. [ABSTRACT FROM AUTHOR]

In this paper, the initial boundary value problem for a fractional nonlocal semilinear pseudo-parabolic equation is established. Firstly, we get the local solution by the standard Galerkin method and the priori estimates. Next, by applying potential well argument, the existence and uniqueness of the global solution are proved for initial energy J (u 0) ≤ d. [ABSTRACT FROM AUTHOR]

In this study, we introduce a numerical method to reconstruct, from posterior measurements of the solution, the initial condition in the one-dimensional heat conduction problem. The method uses a wavelet-based Galerkin method for the spatial discretization and spectral decomposition of the basis stiffness matrix to get the numerical solution without time discretization as in classical approaches. In fact, according to the proposed method, setting an error bound and properly selecting the eigenvalues of the discretization system, we are able to reconstruct the initial conditions without exceeding the required error. The applicability and computational efficiency of the methods are investigated by solving some numerical examples with toy solutions and the experimental results confirm its accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we analyze supercloseness in an energy norm of a weak Galerkin (WG) method on a Bakhvalov-type mesh for a singularly perturbed two-point boundary value problem. For this aim, a special approximation is designed according to the specific structures of the mesh, the WG finite element space and the WG scheme. More specifically, in the interior of each element, the approximation consists of a Gauß–Lobatto interpolant inside the layer and a Gauß–Radau projection outside the layer. On the boundary of each element, the approximation equals the true solution. Besides, with the help of over-penalization technique inside the layer, we prove uniform supercloseness of order k + 1 for the WG method. Numerical experiments verify the supercloseness result and test the influence of different penalization parameters inside the layer. [ABSTRACT FROM AUTHOR]

A computational algorithm for solving the boundary value diffraction problem for periodic 3D lattices of oriented carbon nanotubes is constructed by using the Galerkin projection method. Based on the characteristic equation the electrodynamic calculation of the complex wave number of the quasi-TEM wave and the effective complex permittivity of the anisotropic nanomaterial as a function of frequency was carried out, using the developed computational algorithm for calculating conductivity matrix Y of autonomous blocks with Floquet channels. [ABSTRACT FROM AUTHOR]

This article focuses on constructing and analyzing a non-symmetric interior penalty Galerkin method (NIPG) on Shishkin mesh for solving singularly perturbed 2D elliptic boundary-value problems (BVPs). We use piecewise Lagrange interpolation at Gaussian points to improve the order of convergence of the interpolation error. We then study the superconvergence properties of the NIPG method and prove O (N − 1 ln ⁡ N) k + 1 order of convergence in the discrete energy–norm. Various numerical experiments are provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form − u(4) = f(x, u). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the L2-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order p + 1 in the L2-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most p and mesh size h are used. We then show that the UWLDG solutions are superconvergent with order p + 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method. [ABSTRACT FROM AUTHOR]

This paper uses the Galerkin method to find approximate solutions of some boundary value problems. The solving process requires to solve a system of algebraic equations, which are large and difficult to be solved. According to the Groebner bases theory, an improved Buchberger's algorithm is proposed to solve the algebraic system. The results show that the Galerkin approach is simple and efficient. [ABSTRACT FROM AUTHOR]

Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating boundary value problems using dynamical operators with a Galerkin expansion. A specific choice of basis set in the expansion corresponds to the estimation of dynamical quantities using a Markov state model. More generally, the boundary conditions impose restrictions on the choice of basis sets. We demonstrate how an alternative basis can be constructed using ideas from diffusion maps. In our numerical experiments, this basis gives results of comparable or better accuracy to Markov state models. Additionally, we show that delay embedding can reduce the information lost when projecting the system's dynamics for model construction; this improves estimates of dynamical statistics considerably over the standard practice of increasing the lag time. [ABSTRACT FROM AUTHOR]

This article deals with the solvability of the boundary-value problem for the Navier-Stokes equations with a direction-dependent Navier type slip boundary condition in a bounded domain. Such problems arise when steady flows of fluids in domains with rough boundaries are approximated as flows in domains with smooth boundaries. It is proved by means of the Galerkin method that the boundary-value problem has a unique weak solution when the body force and the variability of the surface friction are sufficiently small compared to the viscosity and the surface friction. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the quaternary classical continuous optimal control for a quaternary linear parabolic boundary value problems(QLPBVPs). The existence and uniqueness theorem of the continuous quaternary state vector solution for the weak form of the QLPBVPs with given quaternary classical continuous control vector (QCCCV) is stated and proved via the Galerkin Method. In addition, the existence theorem of a quaternary classical continuous optimal control vector governinig by the QLPBVPs is stated and demonstrated. The Fréchet derivative for the cost function is derived. Finally, the necessary conditions for the optimality theorem of the proposed problem is stated and demonstrated. [ABSTRACT FROM AUTHOR]

We propose and rigorously analyze a semi- and fully discrete discontinuous Galerkin method for an initial and boundary value problem describing inertial viscoelasticity in terms of elastic and viscoelastic stress components and with mixed boundary conditions. The arbitrary-order spatial discretization imposes strongly the symmetry of the stress tensor, and it is combined with a Newmark trapezoidal rule as a time-advancing scheme. We establish stability and convergence properties, and the theoretical findings are further confirmed via illustrative numerical simulations in 2 and 3 dimensions. [ABSTRACT FROM AUTHOR]

This paper deals with the initial boundary value problem for the double dispersion equation with nonlinear damped term and exponential growth nonlinearity in two space dimensions. We first establish the local well-posedness in the natural energy space by the standard Galërkin method and contraction mapping principle. Then, we prove the solution is global in time by taking the initial data inside the potential well and the solution blows up in finite time as the initial data in the unstable set. Moreover, finite time blow-up results are provided for negative initial energy and for arbitrary positive initial energy respectively. [ABSTRACT FROM AUTHOR]

In this research article Hermite wavelet based Galerkin method is developed for the numerical solution of Volterra integro-differential equations in onedimension with initial and boundary conditions. These equations include the partial differential of an unknown function and the integral term containing the unknown function which is the memory of the problem. Wavelet analysis is a recently developed mathematical tool in applied mathematics. For this purpose, Hermit wavelet Galerkin method has proven a very powerful numerical technique for the stable and accurate solution of giving boundary value problem. The theorem of convergence analysis and compare some numerical examples with the use of the proposed method and the exact solutions shows the efficiency and high accuracy of the proposed method. Several figures are plotted to establish the error analysis of the approach presented. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematics, Mechanics & Computer Science is the property of Al-Farabi Kazakh National University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Using the method of autonomous blocks with Floquet channels (FAB), the boundary value problem of diffraction for metasurfaces made of rectangular graphene nanoribbons is solved. The descriptor (conductivity matrix Y) of the FAB (metasurface unit cell) is determined from the solution of the diffraction problem by the Galerkin projection method. The results of calculating the frequency dependences of transmittance of TEM-wave through metasurfaces with changes in the structure period, chemical potential, orientation angle of the electric field vector of the incident TEM-wave to graphene nanoribbons, as well as polarization characteristics in the THz frequency range were obtained. [ABSTRACT FROM AUTHOR]

We consider a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of a nonlinear fourth order elliptic boundary value problem of p-biharmonic type and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a discretization of the corresponding minimization problem involving a suitably defined reconstruction operator. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W2,p norm in terms of the associated primal and dual energy functionals. It requires the construction of an equilibrated flux and an equilibrated moment tensor based on a three-field formulation of the C0IPDG approximation. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach. [ABSTRACT FROM AUTHOR]

We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximation, for which we derived an a priori error estimate for our adaptive method and showed its exponential convergence in the number of iterations. In this work, we show the exponential convergence in the number of iterations of the a priori error estimate obtained from the piecewise collocation method, provided that a good estimate of the exact solution of the ordinary differential equation at the Sinc points exists. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on regular and stiff ordinary differential equations. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematics, Mechanics & Computer Science is the property of Al-Farabi Kazakh National University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider an application of the barycentric method for the numerical solution of boundary value problems of mathematical physics. The main assumptions are that the system of partial differential equations of the boundary value problem is solvable in the approximation of the Galerkin method and the boundary of the domain of analysis is piecewise linear. A distinctive feature of the barycentric method is the formation of a global system of basis functions for the area of analysis via barycentric coordinates. Solutions for determining barycentric coordinates are given. A comparison is made of the rate of convergence of the barycentric method and grid methods when solving some typical boundary value problems of mathematical physics. [ABSTRACT FROM AUTHOR]

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix. The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities, including transcendental ones, in which the discretization process is as simple as that in solving linear problems, and only common two-term connection coefficients are needed. All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton's method, which does not require numerical integration in the resulting nonlinear discrete system. The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers. The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids, and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids. In addition, Newton's method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method, including the initial guess far from real solutions. [ABSTRACT FROM AUTHOR]

Conventional finite element methods are known to be tedious in adaptive refinements due to their conformal regularity requirements. Further, the enrichment functions for adaptive refinements are often not readily available in general applications. This work introduces a novel neural network-enriched Partition of Unity (NN-PU) approach for solving boundary value problems via artificial neural networks with a potential energy-based loss function minimization. The flexibility and adaptivity of the NN function space are utilized to capture complex solution patterns that the conventional Galerkin methods fail to capture. The NN enrichment is constructed by combining pre-trained feature-encoded NN blocks with an additional untrained NN block. The pre-trained NN blocks learn specific local features during the offline stage, enabling efficient enrichment of the approximation space during the online stage through the Ritz-type energy minimization. The NN enrichment is introduced under the Partition of Unity (PU) framework, ensuring convergence of the proposed method. The proposed NN-PU approximation and feature-encoded transfer learning form an adaptive approximation framework, termed the neural-refinement (n-refinement), for solving boundary value problems. Demonstrated by solving various elasticity problems, the proposed method offers accurate solutions while notably reducing the computational cost compared to the conventional adaptive refinement in the mesh-based methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundary-value problem for the Kuramoto–Sivashinsky equation with homogeneous Neumann conditions. The problem on the existence and stability of second-kind equilibrium states was studied in two ways: by the Galerkin method and by methods of the modern theory of infinite-dimensional dynamical systems. Some differences in results obtained are indicated. [ABSTRACT FROM AUTHOR]

The paper investigates a boundary value problem in a rectangular domain for a high even order equation with discontinuous coefficients. A criterion for the uniqueness of a solution is obtained using the spectral method. The solution is built in the form of a Fourier series. When justifying the convergence of the series, the problem of small denominators arises. We obtain sufficient conditions for the convergence of the series. [ABSTRACT FROM AUTHOR]

In this paper, superconvergence approximations of the modified weak Galerkin finite element method for the singularly perturbed two-point elliptic boundary-value problem have been studied. On the piecewise uniform Shishkin mesh, we have established the superconvergence error bounds of O (N - 1 ln N) k + 1 in the discrete energy norm, where k is the degree of polynomials used in the finite element space and N is the number of elements. Stability analyses have been carried out for both singularly perturbed reaction–diffusion and convection-dominated problems. Some numerical examples are presented to support the theoretical findings. Moreover, the numerical experiments show that the proposed method has the superconvergence error bounds of O (N - 1 ln N) 2 k in the discrete L ∞ -norm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate initial boundary value problems for Kirchhoff-type diffusion equations ∂ β t u + M (∥ u ∥ 2 H s 0 (Ω)) (− Δ) s u = γ | u | ρ u + g (t , x) ∂ t β u + M (‖ u ‖ H 0 s (Ω) 2) (− Δ) s u = γ | u | ρ u + g (t , x) with the Caputo time fractional derivatives and fractional Laplacian operators. We establish a new compactness theorem concerning time fractional derivatives. By Galerkin method, let 0 < ρ < 4 s N − 2 s 0 < ρ < 4 s N − 2 s when γ < 0 γ < 0 , and 0 < ρ < min { 4 s N , 2 s N − 2 s } 0 < ρ < min { 4 s N , 2 s N − 2 s } when γ > 0 γ > 0 , then we obtain the global existence and uniqueness of weak solutions for Kirchhoff problems. Furthermore, we get the decay properties of weak solutions in L 2 (Ω) L 2 (Ω) and L ρ + 2 (Ω) L ρ + 2 (Ω). Remarkably, the decay rate differs from that in the case β = 1 β = 1. [ABSTRACT FROM AUTHOR]

This paper concerns with the state and proof the existence and uniqueness theorem of triple state vector solution (TSVS) for the triple nonlinear parabolic partial differential equations (TNPPDEs),and triple state vector equations (TSVEs), under suitable assumptions. when the continuous classical triple control vector (CCTCV) is given by using the method of Galerkin (MGA). The existence theorem of a continuous classical optimal triple control vector (CCTOCV) for the continuous classical optimal control governing by the TNPPDEs under suitable conditions is proved. [ABSTRACT FROM AUTHOR]