Hamilton, Ritz, and Elastodynamics

1976 ◽  
Vol 43 (4) ◽  
pp. 684-688 ◽  
Author(s):  
C. D. Bailey
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Free Vibration ◽  
Exact Solutions ◽  
Analytical Solutions ◽  
Vibration Modes ◽  
Cantilever Beams ◽  
Transient Motion ◽  
Linear Elastodynamics ◽  
The Law

The theory of Ritz is applied to the equation that Hamilton called the “Law of Varying Action.” Direct analytical solutions are obtained for the transient motion of beams, both conservative and nonconservative. The results achieved are compared to exact solutions obtained by the use of rigorously exact free-vibration modes in the differential equations of Lagrange and to an approximate solution obtained through the application of Gurtin’s principles for linear elastodynamics. A brief discussion of Hamilton’s law and Hamilton’s principle is followed by examples of results for both free-free and cantilever beams with various loadings.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Novel Stability Results for Caputo Fractional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Abdellatif Ben Makhlouf ◽  
El-Sayed El-Hady
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Mortality Rates ◽  
Exact Analytical Solutions ◽  
Scientific Investigations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Stability Results

Modelling some diseases with large mortality rates worldwide, such as COVID-19 and cancer is crucial. Fractional differential equations are being extensively used in such modelling stages. However, exact analytical solutions for the solutions of such kind of equations are not reachable. Therefore, close exact solutions are of interests in many scientific investigations. The theory of stability in the sense of Ulam and Ulam–Hyers–Rassias provides such close exact solutions. So, this study presents stability results of some Caputo fractional differential equations in the sense of Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias. Two examples are introduced at the end to show the validity of our results. In this way, we generalize several recent interesting results.

Free Vibration of a Simply Supported Bar With a Linearly Variable Height of Cross Section

1962 ◽  
Vol 29 (3) ◽  
pp. 497-501 ◽  
Author(s):  
E. Krynicki ◽  
Z. Mazurkiewicz
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Exact Solutions ◽  
Cross Section ◽  
Approximate Solutions ◽  
Variable Cross ◽  
Simply Supported ◽  
Special Cases ◽  
Formal Integration ◽  
Simple Integration

The problem of vibration of nonhomogeneous bars or bars of variable cross section1 leads to differential equations, which are generally unsolvable by formal integration. It is known that functional coefficients occur in these equations, which make it difficult, if not impossible, to obtain exact solutions by simple integration. Several exact solutions obtained for a few special cases and also some interesting approximate solutions are mentioned in the paper.

scholarly journals Analytical solitons for the space-time conformable differential equations using two efficient techniques

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmad Neirameh ◽  
Foroud Parvaneh
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Space Time ◽  
Nonlinear Phenomena ◽  
Approximate Analytical Solutions ◽  
Coupled Burgers Equation ◽  
Mathematics And Physics

AbstractExact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

scholarly journals Numerical Solution of Fuzzy Multiple Hybrid Single Retarded Delay Differential Equations

2019 ◽  
Vol 8 (3) ◽  
pp. 1946-1949
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Solution ◽  
Hybrid Systems ◽  
Delay Differential Equations ◽  
Fuzzy Systems ◽  
Numerical Solutions ◽  
Delay Systems ◽  
Delay Differential

In this Paper, We are combining so many mathematical-cum-engineering topics such as Fuzzy systems, Delay systems and Hybrid Systems under one roof called Numerical Solutions. The fuzzy valued problem was solved numerically and that approximate solution was compared with that of exact solutions. The non fuzzy and fuzzy valued numerical solutions and their graphical illustrations are also provided for the better understanding of the multiple hybrid single retarded delay problems.

Free Vibration Analysis for Step Tubular Structures of Tall Buildings Supported on Elastic Foundation Soil

2012 ◽  
Vol 166-169 ◽  
pp. 194-199
Author(s):  
Zhen Yan Xiao ◽  
Yun Gong ◽  
Yao Qing Gong
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Tall Buildings ◽  
Free Vibration Analysis ◽  
Vibration Modes ◽  
Tubular Structures ◽  
Foundation Soil ◽  
Nodal Lines

A method based on Ordinary Differential Equations (ODE) solver for free vibration analysis of tubular structures of tall buildings is developed, considering the deformation of the foundation soil as well as the interactions between the foundation and soil, by means of a three dimensional model with continuously distributed mass and stiffness. The nodal lines employed to discretize the computational model of the structures are one-variable functions defined on the nodal lines selected by the analyst to describe the dynamic behavior of the model. The unknown functions determined numerically herein are actual vibration modes that can be also recognized as the deformation functions of a set of conceptual structural components. By a Hamiltonian principle, the governing equations of the free vibration analysis can be obtained, which are a set of ordinary differential equations (ODE) of the vibration modes with their corresponding boundary conditions. The desired frequencies and corresponding vibration modes can be obtained by numerically solving the ODEs with boundary conditions. The method is applied to the tubular structures of tall buildings. The results from the illustration example show that the method is rational and powerful for the free vibration analysis of tall buildings.

scholarly journals Study of coupled nonlinear partial differential equations for finding exact analytical solutions

2015 ◽  
Vol 2 (7) ◽  
pp. 140406 ◽  
Author(s):  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
H. Koppelaar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Analytical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Partial Differential ◽  
Exact Analytical Solutions

Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced ( G ′/ G )-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.

scholarly journals Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Seyma Tuluce Demiray ◽  
Hasan Bulut ◽  
Fethi Bin Muhammad Belgacem
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Simulations ◽  
Ordinary Differential Equations ◽  
Analytical Solutions ◽  
Order Derivative ◽  
Two Dimensional ◽  
Transform Method ◽  
Sumudu Transform ◽  
Fractional Type

We make use of the so-called Sumudu transform method (STM), a type of ordinary differential equations with both integer and noninteger order derivative. Firstly, we give the properties of STM, and then we directly apply it to fractional type ordinary differential equations, both homogeneous and inhomogeneous ones. We obtain exact solutions of fractional type ordinary differential equations, both homogeneous and inhomogeneous, by using STM. We present some numerical simulations of the obtained solutions and exhibit two-dimensional graphics by means of Mathematica tools. The method used here is highly efficient, powerful, and confidential tool in terms of finding exact solutions.

scholarly journals Exact Analytical Solutions of Generalized Fifth-Order KdV Equation by the Extended Complex Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Mehvish Fazal Ur Rehman ◽  
Yongyi Gu ◽  
Wenjun Yuan
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Analytical Solutions ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Complex Method ◽  
Exact Analytical Solutions ◽  
Extended Complex ◽  
Fifth Order ◽  
Physical Phenomena

The recently introduced technique, namely, the extended complex method, is used to explore exact solutions for the generalized fifth-order KdV equation. Appropriately, the rational, periodic, and elliptic function solutions are obtained by this technique. The 3D graphs explain the different physical phenomena to the exact solutions of this equation. This idea specifies that the extended complex method can acquire exact solutions of several differential equations in engineering. These results reveal that the extended complex method can be directly and easily used to solve further higher-order nonlinear partial differential equations (NLPDEs). All computer simulations are constructed by maple packages.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

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