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.

scholarly journals Evaluation of hillslope storage with variable width under temporally varied rainfall recharge

2021 ◽  
Author(s):  
Ping-Cheng Hsieh ◽  
Tzu-Ting Huang
Keyword(s):  
Groundwater Flow ◽  
Analytical Solutions ◽  
Water Conservation ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Time Discretization ◽  
Third Order ◽  
Analytical And Numerical Solutions ◽  
Rainfall Recharge ◽  
Integral Transform Technique

Abstract. This study discussed water storage in aquifers of hillslopes under temporally varied rainfall recharge by employing a hillslope-storage equation to simulate groundwater flow. The hillslope width was assumed to vary exponentially to denote the following complex hillslope types: uniform, convergent, and divergent. Both analytical and numerical solutions were acquired for the storage equation with a recharge source. The analytical solution was obtained using an integral transform technique. The numerical solution was obtained using a finite difference method in which the upwind scheme was used for space derivatives and the third-order Runge–Kutta scheme was used for time discretization. The results revealed that hillslope type significantly influences the drains of hillslope storage. Drainage was the fastest for divergent hillslopes and the slowest for convergent hillslopes. The results obtained from analytical solutions require the tuning of a fitting parameter to better describe the groundwater flow. However, a gap existed between the analytical and numerical solutions under the same scenario owing to the different versions of the hillslope-storage equation. The study findings implied that numerical solutions are superior to analytical solutions for the nonlinear hillslope-storage equation, whereas the analytical solutions are better for the linearized hillslope-storage equation. The findings thus can benefit research on and have application in soil and water conservation.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

An Approximate Method for Transient Radial Flow

1962 ◽  
Vol 2 (03) ◽  
pp. 225-256 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Infinite Series ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Numerical Solutions ◽  
Basic Theory ◽  
Variable Pressure ◽  
Flow Problems ◽  
Disturbed Zone ◽  
Approximate Analytical Solutions ◽  
Wide Range

Abstract The basic equations for the flow of gases, compressible liquids and incompressible liquids are derived and the full implications of linearising then discussed. Approximate solutions of these equations are obtained by introducing the concept of a disturbed zone around the well, which expands outwards into the reservoir as fluid is produced. Many important and well-established results are deduced in terms of simple functions rather than the infinite series, or numerical solutions normally associated with these problems. The wide range of application of this approach to transient radial flow problems is illustrated with many examples including; gravity drainage of depletion-type reservoirs; multiple well systems; well interference. Introduction A large number of problems concerning the flow of fluids in oil reservoirs have been solved by both analytical and numerical methods but in almost all cases these solutions have some disadvantages - the analytical ones usually involve rather complex functions (infinite series or infinite integrals) which are difficult to handle, and the numerical ones tend to mask the physical principles underlying the problem. It would seem appropriate, therefore, to try to find approximate analytical solutions to these problems without introducing any further appreciable errors, so that the physical nature of the problem is retained and solutions of comparable accuracy are obtained. One class of problems will be considered in this paper, namely, transient radial flow problems, and it will be shown that approximate analytical solutions of the equations governing radial flow can be obtained, and that these solutions yield comparable results to those calculated numerically and those obtained from "exact" solutions. It will also be shown that the restrictions imposed upon the dependent variable (pressure) are just those which have to be assumed in deriving the usual diffusion-type equations. The method was originally suggested by Guseinov, whopostulated a disturbed zone in the reservoir, the radius of which increases with time, andreplaced the time derivatives in the basic differential equation by its mean value in the disturbed zone. In this paper it is proposed to review the basic theory leading to the equations governing the flow of homogeneous fluids in porous media and to consider the full implications of the approximation introduced in linearising them. The Guseinov-type approximation will then be applied to these equations and the solutions for the flow of compressible and incompressible fluids, and gases in bounded and infinite reservoirs obtained. As an example of the application of this type of approximation, solutions to such problems as production from stratified reservoirs, radial permeability discontinuities; multiple-well systems, and well interference will be given. These solutions agree with many other published results, and in some cases they may be extended to more complex problems without the computational difficulties experienced by other authors. THEORY In order to review the basic theory from a fairly general standpoint it is proposed to limit the idealising assumptions to the minimum necessary for analytical convenience. The assumptions to be made are the following:That the flow is irrotational.That the formation is of constant thickness.Darcy's Law is valid.The formation is saturated with a single homogeneous fluid. SPEJ P. 225^

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.

Aspects of de Wit's Equations Governing the Apparent Spontaneous Yaw of a Ship

1989 ◽  
Vol 42 (1) ◽  
pp. 144-150
Author(s):  
J. O. Flower
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Linearized Equations ◽  
First Order ◽  
Approximate Analytical Solutions

de Wit has produced an analysis of the apparent spontaneous yaw of a ship when undergoing combined rolling and pitching. This analysis produces a set of four first-order simultaneous differential equations which govern the motion. In de Wit the numerical solutions of these equations for a couple of representative examples are given, as well as the corresponding analytical solutions to the linearized equations. In this communication it is shown how two of the four equations can be solved analytically; these solutions can be used to obtain approximate analytical solutions to the remaining two equations.

A Simple Numerical Simulation Method for Lorenz System Families

2013 ◽  
Vol 444-445 ◽  
pp. 786-790
Author(s):  
Cheng Li Zhang ◽  
Yun Zeng
Keyword(s):  
Analytical Solutions ◽  
Lorenz System ◽  
Numerical Solutions ◽  
Simulation Method ◽  
Chen System ◽  
Similar System ◽  
Recursion Method ◽  
Small Segment ◽  
Approximate Analytical Solutions ◽  
Recursion Formulas

Lorenz system families contain Lorenz system, Chen system and Lu system, their accurate analytical solutions are not yet obtained now. The segmenting recursion method was put forward in this paper, the equations of Lorenz system families were reasonably linearized within small segment, the recursion formulas were obtained by solving the approximate analytical solutions within small segment, and all numerical solutions were got by the recursion formulas. The chaotic motion of Lorenz system families were numerically simulated by means of the segmenting recursion method, the simulation results were compared with Runge-Kutta method. The comparative results show that the segmenting recursion method is very effective to numerically simulate Lorenz system families, not only method is simple, programming is easy, but result is accurate. this method is a universal new method to numerically simulate similar system.

Derivation of an approximate analytical solution for understanding the response characteristics of the EBS

2012 ◽  
Vol 1475 ◽  
Author(s):  
T. Ohi ◽  
T. Chiba ◽  
T. Nakagawa ◽  
T. Takase ◽  
T. Nakazawa ◽  
...  
Keyword(s):  
Analytical Solutions ◽  
Safety Assessment ◽  
Numerical Solutions ◽  
Geological Disposal ◽  
One Dimensional ◽  
Response Characteristics ◽  
Excavation Damaged Zone ◽  
Approximate Analytical Solutions ◽  
Target Values ◽  
Damaged Zone

ABSTRACTTo perform a safety assessment for the geological disposal of radioactive waste, it is important to understand the response characteristics of the disposal system. In this study, approximate analytical solutions for steady-state nuclide releases from the engineered barrier system (EBS) of a repository were derived for an orthogonal one-dimensional diffusion model. In these approximate analytical solutions, inventory depletion, decay during migration and the influence of groundwater flow in the excavation damaged zone (EDZ) were considered. These solutions were simplified by the Taylor theorem in order to clearly represent the response characteristics of the EBS. The validity of these solutions was shown by comparison with numerical solutions. The response characteristics of the EBS are useful for identifying target values for important parameters that would have the effect of improving the robustness of system safety. The robustness of the geological disposal system and the reliability of the safety assessment can thus potentially be improved using the approximate analytical solutions.

scholarly journals Analytical solutions for dense, inclined, granular flow over a rigid, bumpy base

2021 ◽  
Vol 249 ◽  
pp. 03039
Author(s):  
James Jenkins ◽  
Diego Berzi
Keyword(s):  
Analytical Solutions ◽  
Granular Flow ◽  
Numerical Solutions ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
Gravitational Flow ◽  
Approximate Analytical Solutions ◽  
The Mean

We first phrase a boundary-value problem for a dense, steady, fully-developed, gravitational flow of identical inelastic spheres over in inclined bumpy base in the absence of sidewalls. We then obtain approximate analytical solutions for the profiles of the solid volume fraction, the strength of the velocity fluctuations, and the mean velocity of the flow. We compare these with those obtained in numerical solutions of the exact equations.

scholarly journals Nonlinear Dynamical Analysis on Four Semi-Active Dynamic Vibration Absorbers with Time Delay

2013 ◽  
Vol 20 (4) ◽  
pp. 649-663 ◽  
Author(s):  
Yongjun Shen ◽  
Mehdi Ahmadian
Keyword(s):  
Time Delay ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Dynamical Analysis ◽  
Control Procedure ◽  
Vibration Absorbers ◽  
Nonlinear Dynamical ◽  
Dynamic Vibration ◽  
Approximate Analytical Solutions ◽  
Dynamic Vibration Absorbers

In this paper four semi-active dynamic vibration absorbers (DVAs) are analytically studied, where the time delay induced by measurement and execution in control procedure is included in the system. The first-order approximate analytical solutions of the four semi-active DVAs are established by the averaging method, based on the illustrated phase difference of the motion parameters. The comparisons between the analytical and the numerical solutions are carried out, which verify the correctness and satisfactory precision of the approximate analytical solutions. Then the effects of the time delay on the dynamical responses are analyzed, and it is found that the stability conditions for the steady-state responses of the primary systems are all periodic functions of time delay, with the same period as the excitation one. At last the effects of time delay on control performance are discussed.

Electro-osmosis of superimposed fluids in the presence of modulated charged surfaces in narrow confinements

2015 ◽  
Vol 776 ◽  
pp. 390-429 ◽  
Author(s):  
Shubhadeep Mandal ◽  
Uddipta Ghosh ◽  
Aditya Bandopadhyay ◽  
Suman Chakraborty
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Surface Charges ◽  
Charged Surfaces ◽  
Physico Chemical ◽  
Approximate Analytical Solutions ◽  
Electro Osmosis ◽  
Flow Features ◽  
Field Formalism ◽  
Modulation Wavelength

In the present study, we attempt to analyse the electro-osmotic flow of two superimposed fluids through narrow confinements in the presence of axially modulated surface charges. We attempt to solve for the flow structure as well as the interface deformation by both analytical and numerical techniques. Approximate analytical solutions are obtained through asymptotic analysis for low deformations, whereas numerical solutions are obtained by applying the phase field formalism; the numerical solutions are obtained for small as well as large interfacial deformations. The analytical solutions are derived only for the transient deformation of the interface, neglecting the transience in the flow, i.e. the flow is assumed to be quasisteady. The numerical solutions, however, are derived including the effects of inertia and transients in the flow. We attempt to compare our analytical and numerical results and explore the effects of several physico-chemical parameters on the deformation of the interface as well as the nature of the flow. Our analysis reveals that parameters such as the modulation wavelength, surface tension (described through the capillary number), viscosity ratio, permittivity ratio and extent of asymmetry in the potential on the two walls are the major contributors to the deformation and the resulting flow features.

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