Convergence Of Approximate Solutions Research Articles

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equations (FSIDEs) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of approximate solutions, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators, and fixed point approach. Furthermore, we examine the convergence of the Faedo–Galerkin (F–G) approximate solutions to the mild solution of our given problem. Finally, a concrete example involving a partial differential equation is provided to validate the main abstract results.

Numerical Investigation with Convergence and Stability Analyses of Integro-Differential Equations of Second Kind

In this paper, integro-differential equations are solved by using an efficient numerical technique, namely, Multistage Optimal Homotopy Asymptotic method. The existence and uniqueness of solutions are established by the application of Lipschitz condition. Convergence of approximate solutions along with stability are also carried out. Some examples are solved to highlight the vital characteristics of the applied numerical scheme. Error estimation and comparison of derived results with existing exact solutions and those results which already available in the literature through graphical illustrations and tables reveal that Multistage Optimal Homotopy Asymptotic algorithm is more efficient and fruitful.

Approximate solutions for neutral stochastic fractional differential equations

This study focuses on a class of neutral stochastic fractional differential equations of order α∈(1,2] in a separable Hilbert space. The existence and uniqueness of approximate solutions are demonstrated using semigroup theory of bounded linear operators, stochastic analysis techniques, and the Banach contraction principle. The convergence of approximate solutions is illustrated using Faedo–Galerkin approximations. Finally, we give an example to illustrate the abstract results.

Weak solutions to the Cauchy problem of the time-dependent Thomas–Fermi equations

In this paper, we are concerned with the existence of weak solutions of the time-dependent Thomas–Fermi equations. We derive approximate solutions by the fractional step Lax–Friedrichs scheme and establish uniform boundedness of approximate solutions. Based on the uniform energy-type estimates, we establish that the entropy dissipation measures of the weak solution of the one-dimensional time-dependent Thomas–Fermi equations for weak entropy–entropy flux pairs, generated by compactly supported C0∞ test functions, are confined in a compact set in Hloc−1. We prove that the Young measure must be a Dirac measure by the Tartar–Murat commutator relation. The convergence of approximate solutions is established by using the compensated compactness method.

О приближенном решении интегральных уравнений Фредгольма методом коллокационных рациональных сплайн-функций

For arbitrary grids of nodes, an approximate solution of the integral equation Fredholm of the second kind in the form of a collocation rational spline function is obtained. Estimates of the rate of uniform convergence of approximate solutions to the exact solution from the smoothness class $C^r$ for $r=0,1,2$ are presented.

Convergence of approximate solutions by heat kernel for transport-diffusion equation in a half-plane

In this paper, by using the heat kernel and the transport operator on each step of time discretization, approximate solutions for the transport-diffusion equation on the half-plane+2are constructed, and their convergence to a function which satisfies the transport-diffusion equation and the initial and boundary conditions is proved. These approximate solutions can be considered as a deterministic version of (the approximation of) the stochastic representation of the solution to parabolic equation, realized by the relationship between the heat kernel and the Brownian motion. But as they are defined only by an integral operator and transport, their properties and their convergence are proved without using probabilistic notions. The result of this paper generalizes that of recent papers about the convergence of analogous approximate solutions on the whole spacen. In case of the half-plane, it is necessary to elaborate (not trivial) estimates of the smoothness of the approximate solutions influenced by boundary condition.

Hybrid absorbing boundary conditions of PML and CRBC

In this paper we introduce a hybrid absorbing boundary condition (HABC) by combining perfectly matched layer (PML) and complete radiation boundary condition (CRBC) for solving a one-dimensional diffraction grating problem. The new boundary condition is devised in such a way that it can enjoy relative advantages from both methods. The well-posedness of the problem with HABC and the convergence of approximate solutions will be analyzed. Numerical examples to illustrate the efficiency of HABC are also presented.

ANALYTICAL SOLUTION FOR NONLINEAR INTERACTION OF EULER BEAM RESTING ON A TENSIONLESS SOIL

The nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. Exact analytical solutions of the challenging problem are rather complicated. The basic obstacle is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution although being needed to get the solution itself. In the current work, solutions are derived using the approximate Rayleigh-Ritz method. The principal of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied implicitly. Results are presented for various cases, including clamped support and free end condition. Various distributed loading conditions are analyzed. Exact solutions are revised briefly. Accurate high order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. The convergence of approximate solutions towards the exact solutions is verified. For each case detailed results of deflection, bending moment and shear are presented.

Pointwise residual method for solving primal and dual ill-posed linear programming problems with approximate data

We propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved. doi:10.1017/S1446181120000243

Numerical solution of the integral equations of the electromagnetic field in geosteering problems

Extra-deep azimuthal resistivity measurements improve the depth of investigation up to 30 m from the wellbore. Interpretation of electromagnetic logging data in the neighbourhood of a well becomes an important technical problem. We present an efficient parallel method for computation of induction tool responses with multiple transmitter–receiver configurations in 2D pixel-based anisotropic model crossed by an arbitrary well trajectory. The cornerstone of the approach is volume integral equation method. We consider the conductivity distribution as a sum of background and anomalous conductivities. Background conductivity is 1D-layered. Anomalous conductivity has arbitrary 2D distribution. Electromagnetic fields are the superposition of background and anomalous fields. Background fields are calculated exactly using rigorous analytical solution for 1D-layered background model. With this approach, the 2D pixel-based model is treated as an extension of the 1D-layered model. The anomalous fields are required only in pixels with conductivity different from the conductivity of the 1D-layered model. Anomalous fields are calculated using convergent series of integral operators [1]. The approach takes into account conductivity anisotropy and allows obtaining both the exact solution and the fast approximate one. The convergence of approximate solution is investigated on some synthetic examples.

On convergence of approximate solutions to the compressible Euler system

We consider a sequence of approximate solutions to the compressible Euler system admitting uniform energy bounds and/or satisfying the relevant field equations modulo an error vanishing in the asymptotic limit. We show that such a sequence either (i) converges strongly in the energy norm, or (ii) the limit is not a weak solution of the associated Euler system. This is in sharp contrast to the incompressible case, where (oscillatory) approximate solutions may converge weakly to solutions of the Euler system. Our approach leans on identifying a system of differential equations satisfied by the associated turbulent defect measure and showing that it only has a trivial solution.

Convergence of Antiperiodic Boundary Value Problems for First-Order Integro-Differential Equations

In this paper, we investigate the convergence of approximate solutions for a class of first-order integro-differential equations with antiperiodic boundary value conditions. By introducing the definitions of the coupled lower and upper solutions which are different from the former ones and establishing some new comparison principles, the results of the existence and uniqueness of solutions of the problem are given. Finally, we obtain the uniform and rapid convergence of the iterative sequences of approximate solutions via the coupled lower and upper solutions and quasilinearization method. In addition, an example is given to illustrate the feasibility of the method.

POINTWISE RESIDUAL METHOD FOR SOLVING PRIMAL AND DUAL ILL-POSED LINEAR PROGRAMMING PROBLEMS WITH APPROXIMATE DATA

AbstractWe propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved.

Rapid Convergence of Approximate Solutions for Fractional Differential Equations

In this paper, we develop a generalized quasilinearization technique for a class of Caputo’s fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions of order m (m≥0), and we obtain the convergence of the sequences of approximate solutions by establishing the convergence of order k (k≥2).

Entropy solutions to doubly nonlinear integro-differential equations

We consider doubly nonlinear history-dependent problems of the form ∂t[k∗(b(v)−b(v0))]−diva(x,∇v)=f. The kernel k satisfies certain assumptions which are, in particular, satisfied by k(t)=t−αΓ(1−α), i.e., the case of fractional derivatives of order α∈(0,1) is included. We show existence of entropy solutions in the case of a nondecreasing b. An existence result in the case of a strictly increasing b is used to get entropy solutions of approximate problems. Kruzhkov’s method of doubling variables, a comparison principle and the diagonal principle are used to obtain a.e. convergence for approximate solutions. A uniqueness result has been shown in a previous work.

Boundary least squares method with three-dimensional harmonic basis of higher order for solving linear div-curl systems with Dirichlet conditions

Abstract Solving linear divergence-curl system with Dirichlet conditions is reduced to finding an unknown vector function in the space of piecewise-polynomial gradients of harmonic functions. In this approach one can use the boundary least squares method with a harmonic basis of a high order of approximation formulated by the authors previously. The justification of this method is given. The properties of the bilinear form and approximating properties of the basis are investigated. Convergence of approximate solutions is proved. A numerical example with estimates of experimental orders of convergence in $\begin{array}{} {\bf V}_h^p \end{array}$-norm for different parameters h, p (p ⩽ 10) is presented. The method does not require specification of penalty weight function.

Mixed schemes of finite element method for non-standard boundary value problems of the nonlinear theory of thin elastic shells

Variational statements of equilibrium problems for a thin elastic shell within a geometrically nonlinear theory of mean bending for different types of principal boundary conditions, including non-classical, are given. Two classes of shell models are considered: (1) a geometrically nonlinear equilibrium model for an anisotropic shell of a material obeying generalized Hooke’s law; (2) a geometrically and physically nonlinear equilibria model for a shallow shell. Sufficient conditions for their generalized solvability in the corresponding Sobolev spaces are derived. In the case of the non-shallow shell the implicit function theorem is used. In the case of the shallow shell the variation problem is investigated using the generalized Weierstrass principle. Mixed finite element methods based on the use of the second derivatives of the deflection as auxiliary variables are constructed for approximate solutions of these problems. Sufficient conditions for solvability of the corresponding discrete problems are obtained. The convergence of approximate solutions is investigated. Accuracy estimates in the case of sufficiently smooth solutions of the original problems are given. Additional conditions are proposed for the problem on the shallow shell to ensure of implementation of the inequalities of the type of strong monotonicity and Lipschitz-continuity of the differential operator.

Piecewise uniform regularization for the inverse problem of microtomography with a-posteriori error estimate

ABSTRACTAn inverse microtomography problem is under consideration in a class of functions with bounded V H variation. An algorithm for solving this problem is proposed based on Tikhonov's regularization with a special regularizer. The algorithm ensures piecewise uniform convergence of approximate solutions to exact solution of the inverse problem. In addition, the question of a-posteriori error estimate of approximate solutions obtained is considered. A new numerical algorithm for finding this estimate is proposed. Numerical experiments on solving a model inverse problem on the class of functions with bounded V H variation are presented along with the results of a-posteriori error estimate for approximate solutions obtained.

Convergence of Mesh Adaptive Algorithms for Elliptic Singularly Perturbed Boundary Value Problems with Exponential Boundary Layer

In this paper we are considering the Galerkin finite element method as applied to self-adjoint singularly perturbed boundary value problems on Bakhvalov and Shishkin meshes. The method of mesh posterior adaptation in the case of an unknown boundary layer border is analyzed. We have formulated the theorem on the convergence of approximate solutions and the mesh convergence theorem as applied to the extremum partitions and conventional ε-uniform estimates of approximate solution errors on extremum partitions. We cite a scheme for proving these theorems and the results of numerical experiments.

Numerical solution and its analysis for a system of hypersingular integral equations

A system of hypersingular integral equations occurs quite naturally in several branches of science and engineering during the formulation of many boundary value problems. The analytical solution is known for the system of dominant equations. However, there are many real world problems such as crack problems occur in the field of fracture mechanics which may not be formulated as system of dominant equations. Therefore, we propose a numerical method to find the approximate solution for such generalized form. The convergence of the proposed method is proved in L N 2 space. This convergence helps to derive theoretical error bound for the error between the exact and the approximate solution. An application of the proposed method in finding numerical solution of hypersingular integral equation over the curves is shown. Finally, the derived theoretical error bound is numerically calculated and validated with the help of numerical examples. • A numerical method is proposed to find solution for the system of HSIE. • The error bound is derived and convergence of approximate solutions is proved. • All derived theoretical results are validated through test examples.