Galerkin Approximation Of Solution 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.

Vibration frequency and mode localization characteristics of strain gradient variable-thickness microplates

Based on the modified strain gradient theory and Kirchhoff–Love assumptions, a free vibration model is established for variable-thickness microplates with three material length scale parameters, and the corresponding Euler–Lagrange equation is derived. Keeping the microplate volume constant, three kinds of thickness variations along the X-direction are considered: linear, parabolic, and cubic variations. Further, a C2-type four-node 36-degree-of-freedom variable-thickness differential quadrature finite element is developed to solve the resulting variable-coefficient boundary value problem by combining the Gauss–Lobatto and differential quadrature rules. Galerkin approximate solution for fully simply supported microplates is provided as a benchmark for numerical method verification. The convergence and accuracy of the model are verified through several examples. Finally, the vibration frequencies of the microplates under different thickness variation types, taper ratios, thickness ratios, material length scale parameters, and boundary conditions are examined through parametric analysis. Also, the influence of each factor on the mode localization of the microplates is quantitatively characterized by the modal assurance criterion (MAC) for the first time. The numerical results show that the thickness variation has a minor effect on the vibration frequency but a significant influence on the mode shape when the volume of the microplate is constant. The strain gradient, taper ratio, and discontinuous boundary have a remarkable effect on the distribution of higher-order mode shape contour lines and the regions where vibration localization occurs.

Convergence analysis and numerical implementation of projection methods for solving classical and fractional Volterra integro-differential equations

In this article, we discuss the convergence analysis of the classical first-order and fractional-order Volterra integro-differential equations of the second kind with a smooth kernel by reducing them into a system of fractional Fredholm integro-differential equations (FFIDEs). For that, we first reformulate the given equation into a system of fractional Volterra integro-differential equations and then transform it into the system of FFIDEs using a simple transformation. We develop a general framework of the newly defined iterated Galerkin method for the reduced system of equations and investigate the existence and uniqueness of the approximate solutions in the given Banach space. We provide the error estimates and convergence analysis for the iterated Galerkin approximate solutions in the supremum norm without any limiting conditions. Further, we provide the superconvergence results for classical first-order and fractional-order Volterra integro-differential equations by proposing a general framework of multi-Galerkin and iterated multi-Galerkin methods for the reduced system of equations. Moreover, we prove that the order of convergence of the proposed methods increases theoretically and numerically with the increasing order of the fractional derivatives. Finally, numerical implementations and illustrative examples are provided to demonstrate our theoretical aspects.

Faedo–Galerkin method for impulsive second-order stochastic integro-differential systems

This paper studies impulsive second-order stochastic differential systems in a separable Hilbert space X. By using the projection operators, we restrict the given problem to a finite-dimensional subspace. The existence and convergence of estimated solutions for the considered problem are investigated via the theories of cosine family and fractional powers of a closed linear operator. We also examine the existence and convergence of the Faedo–Galerkin approximate solutions. At last, we are constructed some examples to demonstrate the effectiveness of the obtained results.

Error analysis of reiterated projection methods for Hammerstein integral equations

In this paper, we consider the Newton-iteration scheme based on Galerkin and multi-Galerkin operators to solve non-linear integral equations of the Fredholm-Hammerstein type for both smooth and weakly singular algebraic and logarithmic type kernels. By choosing as space of piecewise polynomials subspace of degree , we show that for a smooth kernel, Galerkin and multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders and , respectively, where h is the norm of the partition. For weakly singular kernels, we show that the Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders , for algebraic kernels and for logarithmic kernels. Also the multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders for algebraic kernels and for logarithmic kernels. Numerical results are given for the justification of our theoretical results.

Finite element solution of mass transfer effects on unsteady hydromagnetic convective flow past a vertical porous plate in a porous medium with heat source

The objective of this chapter is to analyze the effect of mass transfer on unsteady hydromagnetic free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate in presence of constant suction and heat source. The governing equations of the flow field are solved using Galerkin finite element method and approximate solutions are obtained for velocity field, temperature field, concentration distribution, skin friction and the rate of heat and mass transfer. The numerical results for some special cases were compared with Das et al . [7] and were found to be in good agreement. The effects of the flow parameters such as Hartmann number ( M ), Grashof number for heat and mass transfer ( Gr , Gc) , Permeability parameter ( Kp ), Schmidt number ( Sc ), Heat source parameter ( S ), Prandtl number ( Pr ) and Eckert number ( Ec ) on the flow field are analyzed with the help of figures. The problem has some relevance in the geophysical and astrophysical studies.

Approximation methods for system of linear Fredholm integral equations of second kind

In this paper, Galerkin, multi-Galerkin methods and their iterated versions are developed for solving the system of linear Fredholm integral equations of the second kind for both smooth and weakly singular algebraic and logarithmic type kernels. Here first we develop the theoretical framework for Galerkin and iterated Galerkin methods to solve the system of linear second kind Fredholm integral equations using piecewise polynomials as basis functions and then obtain the superconvergence results similar to that of single linear Fredholm integral equation of the second kind. We show that iterated Galerkin approximation yields better convergence rates than Galerkin approximate solution. Further we enhance these results by using multi-Galerkin and iterated-multi-Galerkin methods and show that the iterated multi-Galerkin approximation yields improved superconvergence rates over iterated Galerkin and multi-Galerkin approximations. The theoretical results are justified by the numerical results.

Approximation methods for system of nonlinear Fredholm–Hammerstein integral equations

In this article, we apply projection methods and their iterated versions to approximate the solution of system of Fredholm–Hammerstein integral equations with both smooth and weakly singular kernels of algebraic and logarithmic type using the piecewise polynomial basis functions. We show that the iterated Galerkin approximate solution converges to the exact solution faster than the Galerkin approximate solution for both smooth and weakly singular algebraic and logarithmic type kernels. We improve these results further in iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results.

Finite element solution of mass transfer effects on unsteady hydromagnetic convective flow past a vertical porous plate in a porous medium with heat source

The objective of this chapter is to analyze the effect of mass transfer on unsteady hydromagnetic free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate in presence of constant suction and heat source. The governing equations of the flow field are solved using Galerkin finite element method and approximate solutions are obtained for velocity field, temperature field, concentration distribution, skin friction and the rate of heat and mass transfer. The numerical results for some special cases were compared with Das et al . [7] and were found to be in good agreement. The effects of the flow parameters such as Hartmann number ( M ), Grashof number for heat and mass transfer ( Gr , Gc) , Permeability parameter ( Kp ), Schmidt number ( Sc ), Heat source parameter ( S ), Prandtl number ( Pr ) and Eckert number ( Ec ) on the flow field are analyzed with the help of figures. The problem has some relevance in the geophysical and astrophysical studies.

On approximate solutions of the equations of incompressible magnetohydrodynamics

Inspired by an approach proposed previously for the incompressible Navier–Stokes (NS) equations, we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics (MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev space of infinite order, made of C∞ vector fields (with vanishing divergence and mean) on the torus. Given any approximate solution of the MHD Cauchy problem, its a posteriori analysis with the method of the present work allows one to infer a lower bound on the time of existence of the exact solution, and to bound from above the Sobolev distance of any order between the exact and the approximate solution. In certain cases the above mentioned lower bound on the time of existence is found to be infinite, so one infers the global existence of the exact MHD solution. We present some applications of this general scheme; the most sophisticated one lives in dimension d=3, with the ABC flow (perturbed magnetically) as an initial datum, and uses for the Cauchy problem a Galerkin approximate solution in 124 Fourier modes. We illustrate the conclusions arising in this case from the a posteriori analysis of the Galerkin approximant; these include the derivation of global existence of the exact MHD solution with the ABC datum, when the dimensionless viscosity and resistivity are equal and stay above a critical value.

Faedo–Galerkin approximate solutions of a neutral stochastic fractional differential equation with finite delay

In this work, we study a class of neutral stochastic fractional differential equation in an arbitrary separable Hilbert space H. We obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by projection of considered associated integral equation onto finite dimensional space. The existence and uniqueness of solutions to every approximate integral equation are obtained by using Banach fixed point theorems and analytic semigroup theory. We show the convergence of the solutions by using Faedo–Galerkin approximations and give an example to show the effectiveness of the main theory. Finally, we provide the conclusion at the end.

Faedo–Galerkin approximation of second order nonlinear differential equation with deviated argument

In this manuscript, we consider a second order nonlinear differential equation with deviated argument in a separable Hilbert space X. We used the strongly continuous cosine family of linear operators and fixed point method to study the existence of an approximate solution of the second order differential equation. We define the fractional power of the closed linear operator and used it to prove the convergence of the approximate solution. Also, we prove the existence and convergence of the Faedo–Galerkin approximate solution. Finally, we give an example to illustrate the application of these abstract results.

On the Reynolds number expansion for the Navier–Stokes equations

In a previous paper of ours (Morosi and Pizzocchero (2012) [1]) we have considered the incompressible Navier–Stokes (NS) equations on a d-dimensional torus Td, in the functional setting of the Sobolev spaces HΣ0n(Td) of divergence free, zero mean vector fields (n>d/2+1). In the cited work we have presented a general setting for the a posteriori analysis of approximate solutions of the NS Cauchy problem; given any approximate solution ua, this allows to infer a lower bound Tc on the time of existence of the exact solution u and to construct a function Rn such that ‖u(t)−ua(t)‖n⩽Rn(t) for all t∈[0,Tc). In certain cases it is Tc=+∞, so global existence is granted for u. In the present paper the framework of Morosi and Pizzocchero (2012) [1] is applied using as an approximate solution an expansion uN(t)=∑j=0NRjuj(t), where R is the Reynolds number. This allows, amongst else, to derive the global existence of u when R is below some critical value R∗ (increasing with N in the examples that we analyze). After a general discussion about the Reynolds expansion and its a posteriori analysis, we consider the expansions of orders N=1,2,5 in dimension d=3, with the initial datum of Behr, Nečas and Wu (2001) [11]. Computations of order N=5 yield a quantitative improvement of the results previously obtained for this initial datum in Morosi and Pizzocchero (2012) [1], where a Galerkin approximate solution was employed in place of the Reynolds expansion.

On approximate solutions of the incompressible Euler and Navier–Stokes equations

We consider the incompressible Euler or Navier–Stokes (NS) equations on a torus T d , in the functional setting of the Sobolev spaces H Σ 0 n ( T d ) of divergence free, zero mean vector fields on T d , for n ∈ ( d / 2 + 1 , + ∞ ) . We present a general theory of approximate solutions for the Euler/NS Cauchy problem; this allows to infer a lower bound T c on the time of existence of the exact solution u analyzing a posteriori any approximate solution u a , and also to construct a function ℛ n such that ‖ u ( t ) − u a ( t ) ‖ n ⩽ ℛ n ( t ) for all t ∈ [ 0 , T c ) . Both T c and ℛ n are determined solving suitable “control inequalities”, depending on the error of u a ; the fully quantitative implementation of this scheme depends on some previous estimates of ours on the Euler/NS quadratic nonlinearity (Morosi and Pizzocchero (2010, in press) [7,8]). To keep in touch with the existing literature on the subject, our results are compared with a setting for approximate Euler/NS solutions proposed in Chernyshenko et al. (2007) [1]. As a first application of the present framework, we consider the Galerkin approximate solutions of the Euler/NS Cauchy problem, with a specific initial datum considered in Behr et al. (2001) [9]: in this case our methods allow, amongst else, to prove global existence for the NS Cauchy problem when the viscosity is above an explicitly given bound.

A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in ℝ 3 with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier boundary conditions. Then we develop a spectral theory of the Stokes operator acting on divergence-free vector fields on a domain with the kinematic and Navier boundary conditions. Finally, we employ the spectral theory and the necessary estimates to construct the Galerkin approximate solutions and establish their convergence to global weak solutions, as well as local strong solutions, of the initial-boundary value problem. Furthermore, we show as a corollary that, when the slip length tends to zero, the weak solutions constructed converge to a solution to the incompressible Navier-Stokes equations subject to the no-slip boundary condition for almost all time. The inviscid limit of the strong solutions to the unique solution of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.

Faedo–Galerkin approximate solutions for stochastic semilinear integrodifferential equations

In this paper, we consider a class of stochastic semilinear integrodifferential equations and prove the existence, uniqueness and approximate solutions in a separable Hilbert space. The convergence of solutions using Faedo–Galerkin approximations is established. For illustration, an example is worked out.

ON APPROXIMATE SOLUTIONS OF SEMILINEAR EVOLUTION EQUATIONS II: GENERALIZATIONS, AND APPLICATIONS TO NAVIER–STOKES EQUATIONS

In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus Tdof any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces ℍn(Td), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e. giving the values of all the necessary constants; this makes a difference with most of the previous literature). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their ℍndistance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).

Enhanced Coercivity for Pure Advection and Advection–Diffusion Problems

We present a common framework in which to set advection problems or advection---diffusion problems in the advection dominated regime, prior to any discretization. It allows one to obtain, in an easy way via enhanced coercivity, a bound on the advection derivative of the solution in a fractional norm of order ?1/2. The same bound trivially applies to any Galerkin approximate solution, yielding a stability estimate which is uniform with respect to the diffusion parameter. The proposed formulation is discussed within Fourier methods and multilevel (wavelet) methods, for both steady and unsteady problems.

Fugue effect on bending of nonlinear viscoelastic beams

Based on the Leaderman constitutive relations in nonlinear viscoelasticity and the linear geometrical assumption, a mathematical model for the bending of nonlinear viscoelastic beams was established in this paper. The Laplace transformation method and the Titchmarsh theorem were used to prove that some relations exist between solutions to bending problems of visco- and elastic beams, which reveals the fugue effect of viscoelastic materials. The high-order Galerkin approximate solution to the quasi-static response of nonlinear viscoelastic beams under a step load was obtained by using the new method suggested in this paper as well as the Mathematica software and the Newton iteration technique.

INERTIAL ALGORITHMS FOR THE STATIONARY NAVIER-STOKES EQUATIONS

Several kind of new numerical schemes for the stationary Navier-Stokes equations based on the virtue of Inertial Manifold and Approximate Inertial Manifold, which we call them inertial algorithms in this paper, together with their error estimations are presented. All these algorithms are constructed under an uniform frame, that is to construct some kind of new projections for the Sobolev space in which the true solution is sought. It is shown that the proposed inertial algorithms can greatly improve the convergence rate of the standard Galerkin approximate solution with lower computing effort. And some numerical examples are also given to verify results of this paper.