scholarly journals The Existence of Fourier Coefficients and Periodic Multiplicity Based on Initial Values and One-Dimensional Wave Limits Requirements

2020 ◽  
Vol 10 (2) ◽  
pp. 146
Author(s):  
Adi Jufriansah ◽  
Azmi Khusnani ◽  
Arief Hermanto ◽  
Mohammad Toifur ◽  
Erwin Prasetyo
Keyword(s):  
Finite Difference ◽  
Analytical Solution ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Fourier Analysis ◽  
Spectral Method ◽  
Difference Method ◽  
Hyperbolic Model ◽  
Solution Approach ◽  
The Finite Difference Method

Physical systems in partial differential equations can be interpreted in a visual form using a wave simulation. In particular, the interpretation of the differential equations used is in the nonlinear hyperbolic model, but in its completion, there are some limitations to the stability requirements found. The aim of this study is to investigate the analytical and numerical analysis of a wave equation with a similar unit and fractal intervals using the Fourier coefficient. The method in this research is to use the analytical solution approach, the spectral method, and the finite difference method. The hyperbolic wave equation's analytical solution approach, illustrated in the Fourier analysis, uses a pulse triangle. The spectral method minimizes errors when there is the addition of the same sample grid points or the periodic domain's expansion with a trigonometric basis. Meanwhile, different ways offer a more efficient solution. Based on the research results, the information obtained is that the Fourier analysis illustrates the pulse triangle use to solve the solution. These results are also suitable for adding sample points to the same spectra. Fourier analysis requires a relatively long time to solve one pulse triangle graph to need another solution, namely the finite difference method. However, its use is still limited in terms of stability when faced with more complex problems.

scholarly journals A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations

2021 ◽  
Vol 15 ◽  
pp. 174830262110084
Author(s):  
Xianjuan Li ◽  
Yanhui Su
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Spectral Method ◽  
Fractional Differential Equations ◽  
Difference Method ◽  
Spectral Collocation ◽  
Parallel Method ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel method.

scholarly journals Timoshenko Beam Theory:First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Timoshenko Beam ◽  
Difference Method ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Order Analysis ◽  
The Finite Difference Method

This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. The FDM is an approximate method for solving problems described with differential or partial differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness). The introduction of additional points allows satisfying boundary and continuity conditions. First-order, second-order, and vibration analyses of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. In addition, tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis, and vibration analysis). Finally, the direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, considering the damping. The efforts and displacements could be determined at any time.

scholarly journals Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Vibration Analysis ◽  
Mass Density ◽  
Functionally Graded ◽  
Governing Equations ◽  
Mass System ◽  
The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

scholarly journals CALCULATION OF A THREE-LAYER CYLINDRICAL SHELL TAKING THE CREEP INTO ACCOUNT

2019 ◽  
Vol 6 (4) ◽  
pp. 14-18 ◽  
Author(s):  
Антон Чепурненко ◽  
Anton Chepurnenko ◽  
Батыр Языев ◽  
Batyr Yazyev ◽  
Анастасия Лапина ◽  
...  
Keyword(s):  
Cylindrical Shell ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Cylindrical Shells ◽  
Euler Method ◽  
Axisymmetric Loading ◽  
The Finite Difference Method

The article presents the derivation of the resolving equations for the calculation of three-layer cylindrical shells under axisymmetric loading, taking into account creep. The problem is reduced to a system of two ordinary differential equations. The solution is performed numerically using the finite difference method in combination with the Euler method.

Transients During a Railway Air Brake Release Demand

1990 ◽  
Vol 204 (1) ◽  
pp. 31-38 ◽  
Author(s):  
M A Murtaza ◽  
S B L Garg
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Laboratory Data ◽  
One Dimensional ◽  
The Finite Difference Method ◽  
Air Brake System ◽  
Good Agreement

This paper deals with the simulation of railway air brake release demand of a twin-pipe graduated release railway air brake system based on the solution of partial differential equations governing one-dimensional flow by the finite difference method supported by extrapolation/interpolation. Air brake release demand is simulated as an exponential input of pressure. The analysis incorporates the corrections needed to be used for various restrictions in the brake pipeline. Results are in good agreement with the laboratory data.

scholarly journals Mathematical Model of the Processoof Pearlite Austenitization

2014 ◽  
Vol 59 (3) ◽  
pp. 981-986 ◽  
Author(s):  
I. Olejarczyk-Wożeńska ◽  
H. Adrian ◽  
B. Mrzygłód
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Diffusion Mechanism ◽  
System Of Differential Equations ◽  
Ttt Diagram ◽  
Austenite Transformation ◽  
The Finite Difference Method

Abstract The paper presents a mathematical model of the pearlite - austenite transformation. The description of this process uses the diffusion mechanism which takes place between the plates of ferrite and cementite (pearlite) as well as austenite. The process of austenite growth was described by means of a system of differential equations solved with the use of the finite difference method. The developed model was implemented in the environment of Delphi 4. The proprietary program allows for the calculation of the rate and time of the transformation at an assumed temperature as well as to determine the TTT diagram for the assigned temperature range.

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