scholarly journals Corrigendum

1985 ◽  
Vol 34 (3) ◽  
pp. 481-483
Author(s):  
T. G. Forbes ◽  
E. R. Priest ◽  
A. W. Hood
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Current Sheet ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Linear Solution ◽  
Significant Change ◽  
Parameter Values ◽  
A Current

Numerical solutions were obtained by Forbes, Priest & Hood (1982) for the resistive decay of a current sheet in an MHD fluid. To check the accuracy of the numerical solutions, a linear, analytical solution was also deived for the regime where diffusion is dominant. In a subsequent reinvestigation of this problem an error in the linear, analytical solution has been discovered. For the parameter values used in the numerical solution this error is too small (≲ 2%) to produce any significant change in the previous test comparison between the numerical and analytical solutions. However, for parameter values much different from those used in the numerical solution, the error in the linear solution can be significant.

Low-frequency drift-induced instabilities in a magnetized two-ion plasma

1986 ◽  
Vol 35 (3) ◽  
pp. 393-412 ◽  
Author(s):  
R. Bharuthram ◽  
M. A. Hellberg
Keyword(s):  
Dispersion Relation ◽  
Numerical Solutions ◽  
Cyclotron Frequency ◽  
Low Frequency ◽  
Frequency Drift ◽  
Transition Diagram ◽  
Electrostatic Waves ◽  
Ion Plasma ◽  
Parameter Values ◽  
A Current

Numerical solutions of a dispersion relation for low-frequency electrostatic waves in a current-carrying, cold, weakly collisional, magnetized two-ion plasma are used to discuss the two-stream and resistive natures of the ion-ion hybrid instability. An instability with analogous behaviour is found to be associated with the light ion cyclotron frequency. Analytical results explain the behaviour. A numerically derived transition diagram summarizes the parameter values for which transitions between different modes take place.

scholarly journals Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator

2021 ◽  
Vol 5 (3) ◽  
pp. 64
Author(s):  
Igor V. Malyk ◽  
Mykola Gorbatenko ◽  
Arun Chaudhary ◽  
Shivani Sharma ◽  
Ravi Shanker Dubey
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Analytical Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Derivative Operator ◽  
Uniqueness Of The Solution ◽  
Time Fractional Diffusion Equation

In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

Analytical Solutions of Saint Venant Equations Decomposed in Frequency Domain

2004 ◽  
Vol 20 (3) ◽  
pp. 187-197
Author(s):  
W. H. Chung ◽  
Y. L. Kang
Keyword(s):  
Analytical Solution ◽  
Finite Difference Method ◽  
Frequency Domain ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Single Equation ◽  
Response Functions ◽  
Generalized Model ◽  
Saint Venant Equations ◽  
Flow Domain

AbstractThe Saint Venant equations are often merged into a single equation for being easily solvable. By doing so, the most general form of the single equation is formulated in this study if all terms are preserved. As a result, the generalized model (GM) results and contains several unexpected nonlinear terms that may impose a great limitation on model analyses. In order to identify these redundant terms, this paper discusses the employment of the linearized Saint Venant equations (LSVE) governing subcritical flow in prismatic channels. The LSVE is solved by a new procedure that separates, in the Laplace frequency domain, the governing equation of water depth from that of flow velocity and thus enables us to consider two independent equations rather than two coupled ones. This allows us to obtain analytical solutions in a much easier way. Comparisons of the response functions of LSVE and the linearized generalized model (LGM) show that the two equations provide identical solutions if the redundant terms embedded in LGM are neglected. It then follows that the response function of LGM can be utilized as a replacement for solving the analytical solution of LSVE that is valid for prismatic channels of any shape. Validity of the analytical solution is verified by repeatedly comparing with the corresponding numerical solutions of finite difference method or Crump's algorithm [1], depending on whether the flow domain is finite or semi-infinite. It is clearly demonstrated in this paper that LSVE serves as an excellent substitution for LGM whose variants have been employed for quite a few years.

scholarly journals Approximate Solutions to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Numerical Solution

2021 ◽  
Author(s):  
Wedad Albalawi ◽  
Alvaro H. Salas ◽  
S. A. El-Tantawy
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Classical Mechanics ◽  
Elliptic Functions ◽  
Approximate Solutions ◽  
Electronic Circuits ◽  
Chebyshev Collocation ◽  
Pendulum Equation ◽  
First Time

Abstract In this work, novel semi-analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation on the pivot vertically for arbitrary angles are obtained for the first time. The semi-analytical solution is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical analysis, the Chebyshev collocation numerical method is introduced for analyzing tthe forced damped parametric driven pendulum equation. Moreover, the semi-analytical solution and Chebyshev collocation numerical solution are compared with the Runge-Kutta (RK) numerical solution. Also, the maximum distance error to the obtained approximate solutions is estimated with respect to the RK numerical solution. The obtained results help many authors to understand the mechanism of many phenomena related to the plasma physics, classical mechanics, quantum mechanics, optical fiber, and electronic circuits.

Verification of ANSYS and Matlab Conduction Results Using Analytical Solutions

2020 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
Giampaolo D'Alessandro ◽  
James V. Beck
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Early Time ◽  
The Body ◽  
Numerical Code ◽  
Transient Temperature ◽  
Dimensionless Time ◽  
Time Step ◽  
First Case

Abstract A two-dimensional transient thermal conduction problem is examined and numerical solutions to the problem generated by ANSYS and Matlab, employing the finite element method, are compared against an analytical solution. Various different grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ANSYS software, including coarse, medium and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. The errors found in the numerical solutions by comparing them directly with the analytical solution vary depending primarily on the time step size used. The errors are much larger if calculated using the analytical solution at a given time as a basis of the comparison between the two solutions as opposed to using the steady-state temperature as a basis. The largest errors appear in the early time steps of the problem, which is typically the regime wherein the largest errors occur in mathematical solutions to transient conduction problems. Conversely, errors at larger values of dimensionless time are extremely small and the numerical solutions agree within one tenth of one percent of the analytical solutions at even the worst locations. In addition to difficulties during the early time values of the problem, temperatures calculated on convective boundaries or prescribed-heat-flux boundaries are locations generating larger-magnitude errors. Corners are particularly difficult locations and require finer gridding and finer time steps in order to generate a very precise solution from a numerical code. These regions are compared, using several grid densities, against the analytical solutions. The analytical solutions are, in turn, intrinsically verified to eight significant digits by comparing similar analytical solutions against one another at very small values of dimensionless time. The solution developed using the Matlab differential equation solver was found to have errors of a similar magnitude to those generated using ANSYS. Two different test cases are examined for the various numerical solutions using the selected grid densities. The first case involves steady heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case involves constant heating for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times were extremely small, the errors found within the short duration test were more significant.

scholarly journals On the Analytical and Numerical Solutions of the Linear Damped NLSE for Modeling Dissipative Freak Waves and Breathers in Nonlinear and Dispersive Mediums: An Application to a Pair-Ion Plasma

2021 ◽  
Vol 9 ◽  
Author(s):  
S. A. El-Tantawy ◽  
Alvaro H. Salas ◽  
M. R. Alharthi
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Chebyshev Approximation ◽  
Rogue Waves ◽  
Approximation Technique ◽  
Damping Term ◽  
Analytical And Numerical Solutions ◽  
Ion Plasma ◽  
Finite Differences Method

In this work, two approaches are introduced to solve a linear damped nonlinear Schrödinger equation (NLSE) for modeling the dissipative rogue waves (DRWs) and dissipative breathers (DBs). The linear damped NLSE is considered a non-integrable differential equation. Thus, it does not support an explicit analytic solution until now, due to the presence of the linear damping term. Consequently, two accurate solutions will be derived and obtained in detail. The first solution is called a semi-analytical solution while the second is an approximate numerical solution. In the two solutions, the analytical solution of the standard NLSE (i.e., in the absence of the damping term) will be used as the initial solution to solve the linear damped NLSE. With respect to the approximate numerical solution, the moving boundary method (MBM) with the help of the finite differences method (FDM) will be devoted to achieve this purpose. The maximum residual (local and global) errors formula for the semi-analytical solution will be derived and obtained. The numerical values of both maximum residual local and global errors of the semi-analytical solution will be estimated using some physical data. Moreover, the error functions related to the local and global errors of the semi-analytical solution will be evaluated using the nonlinear polynomial based on the Chebyshev approximation technique. Furthermore, a comparison between the approximate analytical and numerical solutions will be carried out to check the accuracy of the two solutions. As a realistic application to some physical results; the obtained solutions will be used to investigate the characteristics of the dissipative rogue waves (DRWs) and dissipative breathers (DBs) in a collisional unmagnetized pair-ion plasma. Finally, this study helps us to interpret and understand the dynamic behavior of modulated structures in various plasma models, fluid mechanics, optical fiber, Bose-Einstein condensate, etc.

THEORETICAL ANALYTICAL METHOD IN PROBLEM SOLUTION OF LIQUID VACUUM FREEZING IN QUIET STAT

2016 ◽  
Vol 2016 (3) ◽  
pp. 123-128
Author(s):  
Игорь Лобанов ◽  
Igor Lobanov
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Numerical Solutions ◽  
Stationary Process ◽  
Problem Solution ◽  
Quiet State ◽  
Stationary Vacuum ◽  
Vacuum Freezing

A generalized closed analytical solution of the problem of a quasi-stationary process in liquid vacuum freezing in a quiet state with regard to the thickness of the frosting layer ξ whereas heretofore numerical solutions of this problem occurred. The advantage of the analytical solutions obtained of the problem of a quasi-stationary vacuum freezing of moisture in a finedispersion state over existing numerical ones consists in the identification of an immanent tie between defining and determined parameters regarding a thickness of the frosting layer ξ. It is also possible to use them directly at the computation without resorting to the help of computers.

scholarly journals THEORETICAL ANALYTICAL SOLUTION OF PROBLEM ON STATIONARY SUBCRITICAL CURRENT OF GASEOUS HEAT CARRIER IN PIPING BIFURCATIONS OF HEAT-EXCHANGE EQUIPMENT

2019 ◽  
Vol 2019 (9) ◽  
pp. 25-35
Author(s):  
Игорь Лобанов ◽  
Igor' Lobanov
Keyword(s):  
Heat Exchange ◽  
Heat Carrier ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Special Functions ◽  
Equation System ◽  
Air Space ◽  
Heat Exchange Equipment ◽  
Transcendental Equations ◽  
A Current

The aim of the paper consists in obtaining analytical solutions of the problem on current parameters in the flow bifurcations of a gaseous heat carrier in tubes of heat-exchange equipment used in air-space, shipbuilding and other engineering. The investigation method consists in the solution of the equation system of momentum, continuity and power. In the paper there is substantiated a choice of a theoretical model for a current simulation of a gaseous heat carrier in piping bifurcations of heat-exchange equipment with the allowable degree of proximity to an actual current and complexity of essential computations – a thermo-dynamic model of a subcritical, stationary current of compressible gas. There are obtained analytical solutions of the problem on the current of gaseous heat carrier flows in piping bifurcations of heat-exchange equipment used in different fields of engineering, in particular, in aero-space and shipbuilding and so on, whereas earlier took place only numerical solutions of this problem. In this paper the solutions were obtained without application of special functions used at the solution of non-linear and transcendental equations.

scholarly journals Long wave generation and coastal amplification due to propagating atmospheric pressure disturbances

2021 ◽  
Vol 106 (2) ◽  
pp. 1195-1221
Author(s):  
Gozde Guney Dogan ◽  
Efim Pelinovsky ◽  
Andrey Zaytsev ◽  
Ayse Duha Metin ◽  
Gulizar Ozyurt Tarakcioglu ◽  
...  
Keyword(s):  
Numerical Model ◽  
Analytical Solution ◽  
Shallow Water ◽  
Analytical Solutions ◽  
Atmospheric Pressure ◽  
Numerical Solutions ◽  
Ocean Waves ◽  
Tsunami Waves ◽  
Long Wave ◽  
Dance Suite

AbstractMeteotsunamis are long waves generated by displacement of a water body due to atmospheric pressure disturbances that have similar spatial and temporal characteristics to landslide tsunamis. NAMI DANCE that solves the nonlinear shallow water equations is a widely used numerical model to simulate tsunami waves generated by seismic origin. Several validation studies showed that it is highly capable of representing the generation, propagation and nearshore amplification processes of tsunami waves, including inundation at complex topography and basin resonance. The new module of NAMI DANCE that uses the atmospheric pressure and wind forcing as the other inputs to simulate meteotsunami events is developed. In this paper, the analytical solution for the generation of ocean waves due to the propagating atmospheric pressure disturbance is obtained. The new version of the code called NAMI DANCE SUITE is validated by comparing its results with those from analytical solutions on the flat bathymetry. It is also shown that the governing equations for long wave generation by atmospheric pressure disturbances in narrow bays and channels can be written similar to the 1D case studied for tsunami generation and how it is integrated into the numerical model. The analytical solution of the linear shallow water model is defined, and results are compared with numerical solutions. A rectangular shaped flat bathymetry is used as the test domain to model the generation and propagation of ocean waves and the development of Proudman resonance due to moving atmospheric pressure disturbances. The simulation results with different ratios of pressure speed to ocean wave speed (Froude numbers) considering sub-critical, critical and super-critical conditions are presented. Fairly well agreements between analytical solutions and numerical solutions are obtained. Additionally, basins with triangular (lateral) and stepwise shelf (longitudinal) cross sections on different slopes are tested. The amplitudes of generated waves at different time steps in each simulation are presented with discussions considering the channel characteristics. These simulations present the capability of NAMI DANCE SUITE to model the effects of bathymetric conditions such as shelf slope and local bathymetry on wave amplification due to moving atmospheric pressure disturbances.

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

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