Existence and uniqueness of weak solutions for nonlocal parabolic problems via the Galerkin method

2018 ◽  
Vol 463 (2) ◽  
pp. 910-921
Author(s):  
Miguel Yangari
Keyword(s):  
Galerkin Method ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Parabolic Problems ◽  
The Galerkin Method

scholarly journals Development of a parallel algorithm based on an implicit scheme for the discontinuous Galerkin method for solving diffusion type equations

2020 ◽  
Vol 22 (1) ◽  
pp. 94-106
Author(s):  
Ruslan V. Zhalnin ◽  
Nikita A. Kuzmin ◽  
Victor F. Masyagin
Keyword(s):  
Galerkin Method ◽  
Parallel Algorithm ◽  
Numerical Study ◽  
Sparse Matrices ◽  
Parabolic Type ◽  
Implicit Scheme ◽  
Basis Functions ◽  
Parabolic Problems ◽  
Diffusion Type ◽  
The Galerkin Method

The paper presents a numerical parallel algorithm based on an implicit scheme for the Galerkin method with discontinuous basis functions for solving diffusion-type equations on triangular grids. To apply the Galerkin method with discontinuous basis functions, the initial equation of parabolic type is transformed to a system of partial differential equations of the first order. To do this, auxiliary variables are introduced, which are the components of the gradient of the desired function. To store sparse matrices and vectors, the CSR format is used in this study. The resulting system is solved numerically using a parallel algorithm based on the Nvidia AmgX library. A numerical study is carried out on the example of solving two-dimensional test parabolic initial-boundary value problems. The presented numerical results show the effectiveness of the proposed algorithm for solving parabolic problems.

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.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

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