An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-fractional Schrdinger and Coupled Schrdinger Systems

The objective of this paper is to solve the timefractional Schr¨odinger and coupled Schr¨odinger differential equations(TFSE) with appropriate initial conditions by using the Haar wavelet approximation. For the most part, this endeavor is made to enlarge the pertinence of the Haar wavelet method to solve a coupled system of time-fractional partial differential equations. As a general rule, piecewise constant approximation of a function at different resolutions is presentational characteristic of Haar wavelet method through which it converts the differential equation into the Sylvester equation that can be further simplified easily. Study of the TFSE is theoretical and experimental research and it also helps in the development of automation science,physics, and engineering as well. Illustratively, several test problems are discussed to draw an effective conclusion, supported by the graphical and tabulated results of included examples, to reveal the proficiency and adaptability of the method.

Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders

The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and at t = 0＋ both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies Ω1 and Ω2. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.

Impact of Seasonal Low on Sea Level Rise

Every region around the globe has its unique climatic conditions which are set based on different orographic constant and atmospheric dynamic features. These features posses’ variability on different time scales. Determining the local sea level change based on terrestrial non-tidal, short-term variability is complicated. Some internal mechanisms of ocean are also taking place along with the external physical ones. We show that variability at Sindh-Baluchistan coastal belt can be greatly explained via dimensional indices of the position and intensity of the atmospheric center of action (COAs). This technique has already proved its usefulness at number of location especially in Northern Atlantic. It takes into account the changes in the atmospheric pressure which is exerted on the sea surface influencing the variability in sea level on seasonal scale and on inter-annual basis. As warming causes thermal expansion of water it also causes changes in atmospheric circulation. Both of these processes affect the sea level variability on their respective time scales. Atmospheric being the quicker one of the two to pass on the effect is also more influential to explain the variability in local sea level. In this attempt the COA approach is used to assess the impact of low pressure on local sea levels.

Impact of North-South Shift of Azores High on Summer Precipitation over North West Europe

Several Studies demonstrate that North Atlantic Oscillation (NAO) influences variability of climate over Europe. As NAO is has significant influence on climate of Europe during boreal cold season (November to April), we use the centers of action approach for the study of summer precipitation (June to August) variability over Europe, taking into account variations in the components of the NAO North Atlantic Oscillation (NAO), the Azores High and the Icelandic Low pressure systems. This study shows that north-south shifts of the Azores High has significant impact on interannual variations of summer precipitation over North West Europe, there being more precipitation when the Azores High shifts southward versus when it is northward. Thus this article demonstrate that when the Azores High system is to the south there is flux of moist and warm air from the Atlantic into NW Europe. We present a regression model for summer precipitation over North-west in which the Azores High latitude and the Icelandic low longitude are independent variables and it explains 53 percent of the variance of precipitation during 1952-2002, a significant enhancement over the NAO value of R2 = 0.10.

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 ＋ , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.

Axial Couette flow of an Oldroyd-B fluid in an annulus

This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid motion are graphically illustrated.

Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law

The problem of the steady, incompressible, three dimensional stagnation point flow of a micropolar fluid over an off centered infinite rotating disk in a porous medium is studied in this article. Injection/suction is applied uniformly throughout the surface of porous disk. The Darcy's resistance for the micropolar fluid is also formulated. The partial differential equations are converted into the set of ordinary differential equation by utilizing the suitable transformation. The system of equations is analytically solved by the means of a non-perturbative technique, homotopy analysis method (HAM). The influence of rotational parameter, material parameter, spin gradient viscosity parameter, micro-inertia density parameter, porosity parameter and suction/injection parameter on velocity functions is presented in graphical form and discussed in detail. Verification of the solutions is made by a numerical comparison with the previous study.

Behavioral effects of a four-wing attractor with circuit realization: a cryptographic perspective on immersion

In this paper,we propose an innovative chaotic system,combining fractional derivative and sinehyperbolic nonlinearity with circuit execution.The study of this system is conducted via an analog circuit simulator,using two anti-parallel semiconductor diodes to provide hyperbolic sine nonlinearity,and to function as operational amplifiers.The multi-stability of the system is also enhanced with the help of multi-equilibrium points for distinct real orders of system.The system explores the generation of a four-wing attractor in different phases,both numerically and electronically.By changing the input parameters of the system,different graphs are generated for current flow in state,phase,and space,to confirm the precision of the fractional order derivatives.A reasonable simulation shows that the deliberate circuit provides effective chaos in terms of speed and accuracy,which is comensurate with the numerical simulation.This nonlinear chaotic behavior is utilized to encrypt sound(.wav),images(.jpg),and animated(.gif)data which are a major requirement for the security of communication systems.The proposed circuit performs chaos and cryptographic tasks with high-effective analog computation,and constitutes a novel approach to this area of research.

Homotopy perturbation aided optimization procedure with applications to oscillatory fractional order nonlinear dynamical systems

This paper presents an approximate solution of nonlinear fractional differential equations(FDEs)that exhibit an oscillatory behavior by using a metaheuristic technique.The solutions of the governing equations are approximated by using homotopy perturbation method(HPM)along with the fractional derivative in the Caputo sense.The designed methodology is based on a weighted series of HPM in conjunction with a nature-inspired algorithm.The idea is instantly fascinated by the researchers on the consequent implementation of nature-inspired learning algorithms such as a Cuckoo search algorithm(CSA).The usage of CSA has accelerated the minimized search path of error to the convergent values of the solution.The validity and accuracy of the proposed technique are ascertained by calculating the approximate solution and the error norms which ensure the convergence of the approximation that can be further increased.The critical analysis is also provided by the numerical simulation of two different test models.Discussion of key points has been determined by the tabulation of numerical values and graphs.Comparative study of the results with known numerical technique is also performed.

A Metaheuristic Approach to Optimize European Call Function with Boundary Conditions

The purpose of this paper is to investigate the pricing European call option valuation problems under the exercise price,maturity,risk-free interest rate,and the volatility function. An advance methodology,Chebyshev simulated annealing neural network(Ch SANN),is enforced for the Black-Scholes(B-S) model with boundary conditions. Our scheme is stable and easy to implement on B-S equation,for arbitrary volatility and arbitrary interest rate values. Also,the comparative results demonstrate that the attained approximate solutions are converging towards the exact solution. The graphical results show that the increasing flow of the European call option as the exponential increase takes place in assets. The presented algorithm can be further applied to other financial models with certain boundary conditions. The algorithm of the method shows that the approach can also be easily employed on time-fractional B-S equation.