An approximate solution by the Galerkin method of a quasilinear equation with a boundary condition containing the time derivative of the unknown function

2021 ◽  
Author(s):  
Alisher Mamatov ◽  
Sayfiddin Bakhramov ◽  
Alibek Narmamatov
Keyword(s):  
Boundary Condition ◽  
Approximate Solution ◽  
Galerkin Method ◽  
Unknown Function ◽  
Quasilinear Equation ◽  
Time Derivative ◽  
The Galerkin Method

scholarly journals Refining the Galerkin method error estimation for parabolic type problem with a boundary condition

2021 ◽  
Vol 304 ◽  
pp. 03019
Author(s):  
Alisher Mamatov ◽  
Xusanboy Narjigitov ◽  
Dilshod Turdibayev ◽  
Jamshidbek Rakhmanov
Keyword(s):  
Boundary Condition ◽  
Approximate Solution ◽  
Galerkin Method ◽  
Elliptic Problem ◽  
Principal Part ◽  
Parabolic Problem ◽  
Time Derivative ◽  
Parabolic Type ◽  
Generalized Solution ◽  
Method Error

The article considers a parabolic-type boundary value problem with a divergent principal part, when the boundary condition contains the time derivative of the required function: { ut−d/dxiai(x,t,u,∇u)+a(x,t,u,∇u)=0,a0ut+ai(x,t,u,∇u)cos(v,xi)=g(x,t,u,),(x,t)∈St, u(x,0)= u0(x), x∈Ω Such nonclassical problems with boundary conditions containing the time derivative of the desired function arise in the study of a number of applied problems, for example, when the surface of a body, whose temperature is the same at all its points, is washed off by a well-mixed liquid, or when a homogeneous isotropic body is placed in the inductor of an induction furnace and an electro-magnetic wave falls on its surface. Such problems have been little studied, therefore, the study of problems of parabolic type, when the boundary condition contains the time derivative of the desired function, is relevant. In this paper, the definition of a generalized solution of the considered problem in the space H˜1,1(QT) is given. This problem is solved by the approximate Bubnov-Galerkin method. The coordinate system is chosen from the space H1(Ω). To determine the coefficients of the approximate solution, the parabolic problem is reduced to a system of ordinary differential equations. The aim of the study is to obtain conditions under which the estimate of the error of the approximate solution in the norm H1(Ω) has order O(hk−1) The paper first explores the auxiliary elliptic problem. When the condition of the ellipticity of the problem is satisfied, inequalities are proposed for the difference of the generalized solution of the considered parabolic problem with a divergent principal part, when the boundary condition contains the time derivative of the desired function and the solution of the auxiliary elliptic problem. Using these estimates, as well as under additional conditions for the coefficients and the function included in the problem under consideration, estimates of the error of the approximate solution of the Bubnov-Galerkin method in the norm H1(Ω) of order O(hk−1) for the considered nonclassical parabolic problem with divergent principal part, when the boundary condition contains the time derivative of the desired function.

Non-Newtonian oscillatory layer flow, approximate solution

1979 ◽  
Vol 44 (10) ◽  
pp. 2908-2914 ◽  
Author(s):  
Ondřej Wein
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Galerkin Method ◽  
Horizontal Plane ◽  
Oscillatory Flow ◽  
Oscillatory Boundary Layer ◽  
Layer Flow ◽  
Pseudoplastic Liquid ◽  
The Galerkin Method

The problem of the oscillatory flow of pseudoplastic liquid in vicinity of the infinitely long horizontal plane is formulated in stresses. For Re i.e. for conditions of oscillatory boundary layer the problem is solved approximately by the Galerkin method.

Wavelets Galerkin Method for the Fractional Subdiffusion Equation

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
M. H. Heydari
Keyword(s):  
Galerkin Method ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Practical Importance ◽  
Time Derivative ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Equivalent Problem ◽  
Subdiffusion Equation ◽  
The Galerkin Method

The time fractional subdiffusion equation (FSDE) as a class of anomalous diffusive systems has obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order 0<α<1. Since analytically solving this problem is often impossible, proposing numerical methods for its solution has practical importance. In this paper, an efficient and accurate Galerkin method based on the Legendre wavelets (LWs) is proposed for solving this equation. The time fractional derivatives are described in the Riemann–Liouville sense. To do this, we first transform the original subdiffusion problem into an equivalent problem with fractional derivatives in the Caputo sense. The LWs and their fractional operational matrix (FOM) of integration together with the Galerkin method are used to transform the problem under consideration into the corresponding linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account, automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

On the Equivalence of the Galerkin and Rayleigh-Ritz Methods

1962 ◽  
Vol 66 (621) ◽  
pp. 592-592 ◽  
Author(s):  
Josef Singer
Keyword(s):  
Approximate Solution ◽  
Potential Energy ◽  
Galerkin Method ◽  
Ritz Method ◽  
Total Potential Energy ◽  
Energy Expression ◽  
Total Potential ◽  
Elasticity Problems ◽  
The Galerkin Method ◽  
Potential Energy Expression

The Galerkin method for the approximate solution of elasticity problems (see e.g. Ref. 1) is usually presented as an alternative to the Rayleigh-Ritz method. The main distinction between the two methods is stated to be that the former begins with an equation of equilibrium, whereas the latter begins with a total potential energy expression. The two methods are not always equivalent, and the conditions for equivalence are given in Duncan's lucid presentation of the method (Refs. 2 and 3, which unfortunately have been out of print for some time). These conditions, however, are generally omitted from discussions of the Galerkin method, and it is the purpose of this note to re-emphasise them in a slightly different form.

scholarly journals A Numerical Algorithm for a Kirchhoff-Type Nonlinear Static Beam

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jemal Peradze
Keyword(s):  
Approximate Solution ◽  
Boundary Value Problem ◽  
Galerkin Method ◽  
Numerical Algorithm ◽  
Boundary Value ◽  
Iteration Process ◽  
Beam Equation ◽  
Nonlinear Static ◽  
The Galerkin Method ◽  
Nonlinear Iteration

A boundary value problem is posed for an integro-differential beam equation. An approximate solution is found using the Galerkin method and the Jacobi nonlinear iteration process. A theorem on the algorithm error is proved.

scholarly journals Approximate Solution of Fractional Nonlinear Partial Differential Equations by the Legendre Multiwavelet Galerkin Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
M. A. Mohamed ◽  
M. Sh. Torky
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Partial Differential Equations ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
The Galerkin Method

The Legendre multiwavelet Galerkin method is adopted to give the approximate solution for the nonlinear fractional partial differential equations (NFPDEs). The Legendre multiwavelet properties are presented. The main characteristic of this approach is using these properties together with the Galerkin method to reduce the NFPDEs to the solution of nonlinear system of algebraic equations. We presented the numerical results and a comparison with the exact solution in the cases when we have an exact solution to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

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