Forced Nonlinear Oscillations of a Semi-Infinite Beam Resting on a Unilateral Elastic Soil: Analytical and Numerical Solutions

2006 ◽  
Vol 2 (2) ◽  
pp. 155-166 ◽  
Author(s):  
Giovanni Lancioni ◽  
Stefano Lenci
Keyword(s):  
Moving Boundary ◽  
Numerical Solutions ◽  
Nonlinear Oscillations ◽  
Analytical And Numerical Solutions ◽  
Approximate Analytical Solutions ◽  
The One ◽  
Beam Motion ◽  
Euler Bernoulli Beam ◽  
Element Algorithm ◽  
Elastic Soil

The dynamics of a semi-infinite Euler–Bernoulli beam on unilateral elastic springs is investigated. The mechanical model is governed by a moving-boundary hyperbolic problem, which cannot be solved in closed form. Therefore, we look for approximated solutions following two different approaches. From the one side, approximate analytical solutions are obtained by means of perturbation techniques. On the other side, numerical solutions are determined by a self-made finite element algorithm. The analytical and numerical solutions are compared with each other, and the effects of the problem nonlinearity on the beam motion are analyzed. In particular, the superharmonics oscillations and the resonances are investigated in depth.

Analytical Approximate Solution of Space-Time Fractional Diffusion Equation with a Moving Boundary Condition

2011 ◽  
Vol 66 (5) ◽  
pp. 281-288 ◽  
Author(s):  
Subir Das ◽  
Rajnesh Kumar ◽  
Praveen Kumar Gupta
Keyword(s):  
Diffusion Equation ◽  
Moving Boundary ◽  
Numerical Solutions ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Approximate Analytic Solution ◽  
Approximate Analytical Solutions ◽  
Moving Boundary Condition ◽  
Time Fractional Diffusion Equation

August 12, 2010 The homotopy perturbation method is used to find an approximate analytic solution of the problem involving a space-time fractional diffusion equation with a moving boundary. This mathematical technique is used to solve the problem which performs extremely well in terms of efficiency and simplicity. Numerical solutions of the problem reveal that only a few iterations are needed to obtain accurate approximate analytical solutions. The results obtained are presented graphically.

Analytical and Numerical Solutions of the Riccati Equation Using the Method of Variation of Parameters. Application to Population Dynamics

2020 ◽  
Vol 15 (10) ◽  
Author(s):  
Orestes Tumbarell Aranda ◽  
Fernando A. Oliveira
Keyword(s):  
Population Dynamics ◽  
Pattern Formation ◽  
Riccati Equation ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Original Equation ◽  
Analytical And Numerical Solutions ◽  
Variation Of Parameters ◽  
Approximate Analytical Solutions ◽  
Independent Variable

Abstract This work presents new approximate analytical solutions for the Riccati equation (RE) resulting from the application of the method of variation of parameters. The original equation is solved using another RE explicitly dependent on the independent variable. The solutions obtained are easy to implement and highly applicable, which is confirmed by solving several examples corresponding to REs whose solution is known, as well as optimizing the method to determine the density of the members that make up a population. In this way, new perspectives are open in the study of the phenomenon of pattern formation.

TESTING POWER-LAW RELAXATION SCENARIOS IN A METASTABLE LIQUID

2012 ◽  
Vol 26 (29) ◽  
pp. 1250146 ◽  
Author(s):  
BHASKAR SEN GUPTA ◽  
SHANKAR P. DAS
Keyword(s):  
Correlation Function ◽  
Power Law ◽  
Hard Sphere ◽  
Numerical Solutions ◽  
Packing Fraction ◽  
Equilibrium Density ◽  
Density Correlation ◽  
Density Correlation Function ◽  
Basic Model ◽  
The One

The renormalized dynamics described by the equations of nonlinear fluctuating hydrodynamics (NFH) treated at one loop order gives rise to the basic model of the mode coupling theory (MCT). We investigate here by analyzing the density correlation function, a crucial prediction of ideal MCT, namely the validity of the multi step relaxation scenario. The equilibrium density correlation function is calculated here from the direct solutions of NFH equations for a hard sphere system. We make first detailed investigation for the robustness of the correlation functions obtained from the numerical solutions by varying the size of the grid. For an optimum choice of grid size we analyze the decay of the density correlation function to identify the multi-step relaxation process. Weak signatures of two step power law relaxation is seen with exponents which do not match predictions from the one loop MCT. For the final relaxation stretched exponential (KWW) behavior is seen and the relaxation time grows with increase of density. But apparent power law divergences indicate a critical packing fraction much higher than the corresponding MCT predictions for a hard sphere fluid.

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