Exact Explicit Analytical Solution Research Articles

Ultraslow Kuznetsov-Ma solitons and Ahkmediev breathers in a cold three-state medium exposed to nanosecond optical pulses

We investigate the formation of the Kuznetsov-Ma solitons and Ahkmediev breathers in a cold Λ-type three-level atomic system that interacts with a probe field of nanosecond pulse duration and a strong continuous-wave driving field via an electromagnetically induced transparency process. Within the framework of the Hirota equation, exact explicit analytical solutions of these breathers are obtained, showing different amplitude and oscillatory characteristics. Numerical simulations confirm the stability of both types of breathers against non-integrable perturbations that are caused by the nonvanishing decay rates of atomic states. We show that both breathers thus generated can propagate at a quite low group velocity.

An Analysis of Modified Emden-Type Equation ẍ + αxẋ + βx<sup>3</sup> = 0: Exact Explicit Analytical Solution, Lagrangian, Hamiltonian for Arbitrary Values of α and β

The modified Emden-type is being investigated by mathematicians as well as physicists for about a century. However, there exist no exact explicit solution of this equation, ẍ + αxẋ + βx3 = 0 for arbitrary values of α and β. In this work, the exact analytical explicit solution of modified Emden-type (MEE) equation is derived for arbitrary values of α and β. The Lagrangian and Hamiltonian of MEE are also worked out. The solution is also utilized to find exact explicit analytical solution of Force-free Duffing oscillator-type equation. And exact explicit analytical solution of two-dimensional Lotka-Volterra System is also worked out.

Exact Analytical Durometer Hardness Scale Interconversion

Abstract Previous work has related Young’s modulus to durometer hardness for any standardized scale. In this paper, we build on this work to solve explicitly and exactly for the hardness in any one standardized durometer hardness scale as a function of the hardness in any other target scale. We find that when the target scale is for a flat indenter, the conversion is algebraic and straightforward. However, when the target scale is for an indenter that is not flat (conical or hemispherical), the exact explicit analytical solution requires a power series inversion, said series involving beta functions and solutions to a set of integer equations. We complete our analysis with two worked examples illustrating the use of our interconversion equations and charts.

Determination of Commercial Silicon Diode (4007IN) Parameters from the I–V and P-V Characteristics

A simple method is presented for determining the four diode parameters of the single exponential model of a silicon commercial diode, namely: Is reverse saturation current, n diode ideality factor, Rs series resistance and Gp shunt conductance. The current-voltage and power-voltage curves of the corresponding commercial diode are calculated and simulated via the exact explicit analytical solutions. By the use of the Lambert W function in its Pade-type approximation and the the FindFit function in the Mathematica software package, the four parameters were extracted. The fitted current-voltage, power-voltage curves and the corresponding experimental data are in good agreement. Also we provided the Pade-type approximation expression for Lambert W function.

The large-time behaviour of coarsening of a particulate assemblage due to Ostwald ripening and coagulation

A theoretical description of the concluding stage of Ostwald ripening (coalescence) with allowance for coagulation of particles is considered. An implicit integral equation for the distribution function is derived for equal coagulation rates of the particles. This equation is analytically solved in the limit of large times. An exact explicit analytical solution for the volume density distribution function depends on the initial conditions and is in agreement with experimental data.

Parameters Extraction of the Au/SnO 2 -Si(n)/Al p-n Junction Solar Cell Using Lambert W Function

The paper present a simple method for the extraction of solar cells parameters with a single diode circuit model from its dark characteristic and considering the series and shunt resistances. In order to extract the four parameters of the solar cell, an optimized technique is presented by using a combination of Lambert W function in its Pade-type approximation and the FindFit function in the Mathematica software package. By using the exact explicit analytical solutions, the current-voltage curves of the Au/SnO2-Si(n)/Al p-n junction solar cell are calculated and simulated. The results have the good agreement between the fitted current-voltage curves and the experimental data of the corresponding solar cell. A closed form expression for the Lambert W function based on a Pade-type approximation is also provided.

A new method for determining the characteristics of solar cells

A new method is proposed to describe the characteristics of solar cell and module in this article. By using the exact explicit analytical solutions, the current–voltage and power–voltage curves of the corresponding modules are calculated and simulated. In order to extract the parameters of the solar cells, an optimized technique is presented by using Lambert W function and polynomial curve fitting. The accuracy of the model is compared with previous methods in other work. Results have the good agreement between the fitted current–voltage curves and the experimental data. Moreover, error and statistical analyses are carried out to illustrate the accuracy of the estimated parameters and procedure suitability.

Quantum decay model with exact explicit analytical solution

A simple decay model is introduced. The model comprises a point potential well, which experiences an abrupt change. Due to the temporal variation, the initial quantum state can either escape from the well or stay localized as a new bound state. The model allows for an exact analytical solution while having the necessary features of a decay process. The results show that the decay is never exponential, as classical dynamics predicts. Moreover, at short times the decay has a fractional power law, which differs from perturbation quantum method predictions. At long times the decay includes oscillations with an envelope that decays algebraically. This is a model where the final state can be either continuous or localized, and that has an exact analytical solution.

New method to extract the model parameters of solar cells from the explicit analytic solutions of their illuminated characteristics

Abstract We present a new method to extract the intrinsic and extrinsic model parameters of illuminated solar cells containing parasitic series resistance and shunt conductance. The method is based on calculating the Co-content function (CC) from the exact explicit analytical solutions of the illuminated current–voltage (I–V) characteristics. The resulting CC is expressed as a purely algebraic function of current and voltage from whose coefficients the intrinsic and extrinsic model parameters are then readily determined by bidimensional fitting. The procedure is illustrated by applying it to experimental and synthetic I–V characteristics and an analysis of the errors is presented.

Heat transfer across a solid–solid interface obtained by machining in a lathe

In this paper the heat transfer across the interface between two cylinders in axial contact for which the surfaces are obtained by machining in a lathe was studied. The macro-contact generated by this process has a spiral shape. The governing equations being difficult to solve for this geometric configuration, the spiral was simulated by multiple concentric circular annular contacts. Thus, the problem is two-dimensional and hence easier to solve. An exact explicit analytical solution was developed in order to calculate the temperature distribution and the thermal contact resistance R c between the two cylinders. The R c is expressed as a function of the ratio of real contact area to the apparent one, and the number of contacts. A simple correlation is also proposed to calculate the R c with high accuracy.

An Exact Explicit Analytical Solution of the Steady-State Temperature in a Half Space Subjected to a Moving Circular Heat Source

So far in the literature, the distribution of stationary temperature over the surface of a half space subjected to a moving circular heat source has been reported in an integral or asymptotic form. In this paper, an exact explicit analytical solution is provided, which allows the determination of the temperatures over the contact area with a very short computational time, regardless of the value of the Peclet number. The solution is based on special functions (Bessel and hypergeometric functions) that are pre-programmed under a formal calculation software (e.g., Maple). The results of the proposed solution are in agreement with the asymptotic models available in the literature.

Stochastic Optimal Bounded Control for a System With the Boltz Cost Function

A single-degree-of-freedom mass-spring system is considered under white-noise excitation. To reduce expected response energy of the system, a bounded control force can be applied and optimal control law is sought for. An exact explicit analytical solution, for the Lagrange cost function, is derived for the Hamilton-Jacobi-Bellman (HJB) equation within a certain "outer" domain. This solution provides boundary conditions for numerical simulation of the HJB equation within the remaining "inner" domain. Comparison of optimal control law, with a previously obtained one for the Mayer cost function, is made. The analytical solution to the HJB equation for the "combined" Boltz cost function is derived for the outer domain as a linear combination of the solutions for the Mayer and Lagrange cost functions. Sensitivity of the Bellman functional to the above optimal control laws is analyzed.

Dark solitons in weakly saturable nonlinear media

Exact explicit analytical solutions of dark (gray and black) solitons to the quintic nonlinear Schr\"odinger equation that includes the first high order nonlinear saturable term are presented. The dark solitons in the weakly saturable self-defocusing media are shown to be stable to a small perturbation (noncollision type) and, as in the case of a Kerr nonlinearity, the two black solitons launched in parallel repel with propagation distance. However, when the two dark solitons are launched towards each other in tilted angles heading for a collision, the solitons in the saturable nonlinear media could not survive through a collision above a critical background intensity (although they survive below that critical intensity value). This is in contrast to the Kerr law nonlinearity.