Search results for: explicit equation

Authors: Lawrence A. Farinola

Abstract:

Four point implicit schemes for the approximation of the first and pure second order derivatives for the solution of the Dirichlet problem for one dimensional Schrodinger equation with respect to the time variable t were constructed. Also, special four-point implicit difference boundary value problems are proposed for the first and pure second derivatives of the solution with respect to the spatial variable x. The Grid method is also applied to the mixed second derivative of the solution of the Linear Schrodinger time-dependent equation. It is assumed that the initial function belongs to the Holder space C⁸⁺ᵃ, 0 < α < 1, the Schrodinger wave function given in the Schrodinger equation is from the Holder space Cₓ,ₜ⁶⁺ᵃ, ³⁺ᵃ/², the boundary functions are from C⁴⁺ᵃ, and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. It is proven that the solution of the proposed difference schemes converges uniformly on the grids of the order O(h²+ k) where h is the step size in x and k is the step size in time. Numerical experiments are illustrated to support the analysis made.

Keywords: approximation of derivatives, finite difference method, Schrödinger equation, uniform error

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Authors: Kamel Al-Khaled

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Reaction-diffusion equations are commonly used in population biology to model the spread of biological species. In this paper, we propose a fractional reaction-diffusion equation, where the classical second derivative diffusion term is replaced by a fractional derivative of order less than two. Based on the symbolic computation system Mathematica, Adomian decomposition method, developed for fractional differential equations, is directly extended to derive explicit and numerical solutions of space fractional reaction-diffusion equations. The fractional derivative is described in the Caputo sense. Finally, the recent appearance of fractional reaction-diffusion equations as models in some fields such as cell biology, chemistry, physics, and finance, makes it necessary to apply the results reported here to some numerical examples.

Keywords: fractional partial differential equations, reaction-diffusion equations, adomian decomposition, biological species

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Authors: E. V. Krishnan

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In this paper, we employ mapping methods to construct exact travelling wave solutions for a modified Korteweg-de Vries equation. We have derived periodic wave solutions in terms of Jacobi elliptic functions, kink solutions and singular wave solutions in terms of hyperbolic functions.

Keywords: travelling wave solutions, Jacobi elliptic functions, solitary wave solutions, Korteweg-de Vries equation

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Authors: Sagardeep Talukdar, Riki Dutta, Gautam Kumar Saharia, Sudipta Nandy

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In non-linear optics, the Fokas-Lenells equation (FLE) is a well-known integrable equation that describes how ultrashort pulses move across the optical fiber. It admits localized wave solutions, just like any other integrable equation. We apply the Hirota bilinearization method to obtain the soliton solution of FLE. The proposed bilinearization makes use of an auxiliary function. We apply the method to FLE with a vanishing boundary condition, that is, to obtain a bright soliton solution. We have obtained bright 1-soliton and 2-soliton solutions and propose a scheme for obtaining an N-soliton solution. We have used an additional parameter that is responsible for the shift in the position of the soliton. Further analysis of the 2-soliton solution is done by asymptotic analysis. In the non-vanishing boundary condition, we obtain the dark 1-soliton solution. We discover that the suggested bilinearization approach, which makes use of the auxiliary function, greatly simplifies the process while still producing the desired outcome. We think that the current analysis will be helpful in understanding how FLE is used in nonlinear optics and other areas of physics.

Keywords: asymptotic analysis, fokas-lenells equation, hirota bilinearization method, soliton

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Authors: Tai-Ping Chang

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In the present study, the effects on chaotic motion of single-walled carbon nanotube (SWCNT) due to the linear and nonlinear damping are investigated. By using the Hamilton’s principle, the nonlinear governing equation of the single-walled carbon nanotube embedded in a matrix is derived. The Galerkin’s method is adopted to simplify the integro-partial differential equation into a nonlinear dimensionless governing equation for the SWCNT, which turns out to be a forced Duffing equation. The variations of the Lyapunov exponents of the SWCNT with damping and harmonic forcing amplitudes are investigated. Based on the computations of the top Lyapunov exponent, it is concluded that the chaotic motion of the SWCNT occurs when the amplitude of the periodic excitation exceeds certain value, besides, the chaotic motion of the SWCNT occurs with small linear damping and tiny nonlinear damping.

Keywords: chaotic motion, damping, Lyapunov exponents, single-walled carbon nanotube

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Authors: Wan-Lin Tsai

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Speech act of requests is an important issue in the field of language learning and teaching because we cannot avoid making requesting in our daily life. This study examined whether or not the subjects who were freshmen and majored in German at Wenzao University of Languages were able to use the linguistic forms which they had learned from their course book Schritte International to make appropriate requests through dialogue completed tasks (DCT). The results revealed that the majority of the subjects were unable to use the forms to make appropriate requests in German due to the lack of explicit instructions. Furthermore, Chinese interference was observed in students' productions. Explicit instructions in speech acts are strongly recommended.

Keywords: Chinese interference, German pragmatics, German teaching, make appropriate requests in German, speech act of requesting

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Authors: P. Srinivas, P. V. N. Prasad

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Since torque ripple is the main cause of noise and vibrations, the performance of Switched Reluctance Motor (SRM) can be improved by minimizing its torque ripple using a novel control technique called Direct Torque Control (DTC). In DTC technique, torque is controlled directly through control of magnitude of the flux and change in speed of the stator flux vector. The flux and torque are maintained within set hysteresis bands. The DTC of SRM is analysed by two methods. In one of the methods, the actual torque is computed by conducting Finite Element Analysis (FEA) on the design specifications of the motor. In the other method, the torque is computed by Simplified Torque Equation. The variation of peak current, average current, torque ripple and speed settling time with Simplified Torque Equation model is compared with FEA based model.

Keywords: direct toque control, simplified torque equation, finite element analysis, torque ripple

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Authors: Abdel-Maksoud Abdel-Kader Soliman

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The General Regularized Long Wave (GRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid-knot points as element shape. Also, we use the Fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm.

Keywords: generalized RLW equation, solitons, quartic b-spline, nonlinear partial differential equations, difference equations

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Authors: Elisha Kyirem

Abstract:

Undoubtedly, climate change is a major global challenge that could threaten the very foundation upon which life on earth is anchored, with its impacts on human mobility attracting the attention of policy makers and researchers. There is an increasing body of literature and case studies suggesting that migration could be a way through which the vulnerable move away from areas exposed to climate extreme events to improve their lives and that of their families. This presents migration as a way through which people voluntarily move to seek opportunities that could help reduce their exposure and avoid danger from climate events. Thus, migration is seen as a proactive adaptation strategy aimed at building resilience and improving livelihoods to enable people to adapt to future changing events. However, there has not been any mathematical equation linking migration and climate change adaptation. Drawing from literature in development studies, this paper develops an equation that seeks to link the relationship between migration and climate change adaptation. The mathematical equation establishes the linkages between migration, resilience, poverty reduction and vulnerability, and these the paper maintains, are the key variables for conceptualizing the migration-climate change adaptation nexus. The paper then tests the validity of the equation using the sustainable livelihood framework and publicly available data on migration and tourism in Ghana.

Keywords: migration, adaptation, climate change, adaptation, poverty reduction

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Authors: Saeed Ghali, Hoda El-Faramawy, Mamdouh Eissa, Michael Mishreky

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Scale-resistant austenitic stainless steel, X45CrNiW 18-9, has been developed, and modified steels produced through partial and total nickel replacement by nitrogen. These modified steels were produced in a 10 kg induction furnace under different nitrogen pressures and were cast into ingots. The produced modified stainless steels were forged, followed by air cooling. The phases of modified stainless steels have been investigated using the Schaeffler diagram, dilatometer, and microstructure observations. Both partial and total replacement of nickel using 0.33-0.50% nitrogen are effective in producing fully austenitic stainless steels. The nitrogen contents were determined and compared with those calculated using the Institute of Metal Science (IMS) equation. The results showed great deviations between the actual nitrogen contents and predicted values through IMS equation. So, an equation has been derived based on chemical composition, pressure, and temperature at 1600oC. [N%] = 0.0078 + 0.0406*X, where X is a function of chemical composition and nitrogen pressure. The derived equation has been used to calculate the nitrogen content of different steels using published data. The results reveal the difficulty of deriving a general equation for the prediction of nitrogen content covering different steel compositions. So, it is necessary to use a narrow composition range.

Keywords: solubility, nitrogen, stainless steel, Schaeffler

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Authors: Azad A. Mohammed, Dunyazad K. Assi, Alan S. Abdulrahman

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Current ACI 318 Code provides two limits for minimum steel ratio for concrete beams. When concrete compressive strength be larger than 31 MPa the limit of √(fc')/4fy usually governs. In this paper shortcomings related to using this limit was fairly discussed and showed that the limit is based on 90% safety factor and was derived based on modulus of rupture equation suitable for concretes of compressive strength lower than 31 MPa. Accordingly, the limit is nor suitable and critical for concretes of higher compressive strength. An alternative equation was proposed for minimum steel ratio of rectangular beams and was found that the proposed limit is accurate for beams of wide range of concrete compressive strength. Shortcomings of the current ACI 318 Code equation and accuracy of the proposed equation were supported by test data obtained from testing six reinforced concrete beams.

Keywords: concrete beam, compressive strength, minimum steel ratio, modulus of rupture

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Authors: Guesh Simretab Gebremedhin, Saumya Rajan Jena

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In this work, a septic B-spline scheme has been used to simplify the process of solving an approximate solution of the generalized Rosenau-regularized long-wave equation (GR-RLWE) with initial boundary conditions. The resulting system of first-order ODEs has dealt with Butcher’s fifth order Runge-Kutta (BFRK) approach without using finite difference techniques for discretizing the time-dependent variables at each time level. Here, no transformation or any kind of linearization technique is employed to tackle the nonlinearity of the equation. Two test problems have been selected for numerical justifications and comparisons with other researchers on the basis of efficiency, accuracy, and results of the two invariants Mᵢ (mass) and Eᵢ (energy) of some motion that has been used to test the conservative properties of the proposed scheme.

Keywords: septic B-spline scheme, Butcher's fifth order Runge-Kutta approach, error norms, generalized Rosenau-RLW equation

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Authors: Jian-Ping Yu, Wen-Xiu Ma, Yong-Li Sun, Chaudry Masood Khalique

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In this paper, a 2d Toda equation is studied, which is a classical integrable system and plays a vital role in mathematics, physics and other areas. New lump-type solution is constructed by using the Hirota bilinear method. One interesting feature of this research is that this lump-type solutions possesses two types of multiple-lump-type waves, which are one- and two-lump-type waves. Moreover, the corresponding 3d plots, density plots and contour plots are given to show the dynamical features of the obtained multiple-lump-type solutions.

Keywords: 2d Toda equation, Hirota bilinear method, Lump-type solution, multiple-lump-type solution

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Authors: Khashayar Kazemzadeh, Mohammad Hanif Dasoomi

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According to the increasing volume of traffic, lane changing plays a crucial role in traffic flow. Lane changing in traffic depends on several factors including road geometrical design, speed, drivers’ behavioral characteristics, etc. A great deal of research has been carried out regarding these fields. Despite of the other significant factors, the drivers’ behavioral characteristics of lane changing has been emphasized in this paper. This paper has predicted the effective equation based on personal characteristics of lane changing by regression models.

Keywords: effective equation, lane changing, drivers’ behavioral characteristics, regression models

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Authors: H. N. Agiza, M. A. Sohaly, M. A. Elfouly

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Parkinson's disease (PD) is a heterogeneous disorder with common age of onset, symptoms, and progression levels. In this paper we will solve analytically the PD model as a non-linear delay differential equation using the steps method. The step method transforms a system of delay differential equations (DDEs) into systems of ordinary differential equations (ODEs). On some numerical examples, the analytical solution will be difficult. So we will approximate the analytical solution using Picard method and Taylor method to ODEs.

Keywords: Parkinson's disease, step method, delay differential equation, two delays

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Authors: A. R. Nezamabadi, M. Veiskarami

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This paper studies free vibration of simply supported functionally graded beams with piezoelectric layers based on the first order shear deformation theory. The Young's modulus of beam is assumed to be graded continuously across the beam thickness. The governing equation is established. Resulting equation is solved using the Euler's equation. The effects of the constituent volume fractions, the influences of applied voltage on the vibration frequency are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data.

Keywords: mechanical buckling, functionally graded beam, first order shear deformation theory, free vibration

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Authors: Somveer Singh, Vineet Kumar Singh

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This work contributes a numerical method based on Legendre wavelet approximation for the treatment of partial integro-differential equation (PIDE). Operational matrices of Legendre wavelets reduce the solution of PIDE into the system of algebraic equations. Some useful results concerning the computational order of convergence and error estimates associated to the suggested scheme are presented. Illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.

Keywords: legendre wavelets, operational matrices, partial integro-differential equation, viscoelasticity

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Authors: Chao-Qing Dai

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Two families of exact travelling solutions for the discrete sine-Gordon equation are constructed based on the variable-coefficient Jacobian elliptic function method and different transformations. When the modulus of Jacobian elliptic function solutions tends to 1, soliton solutions can be obtained. Some soliton solutions degenerate into the known solutions in literatures. Moreover, dynamical properties of exact solutions are investigated. Our analysis and results may have potential values for certain applications in modern nonlinear science and textile engineering.

Keywords: exact solutions, variable-coefficient Jacobian elliptic function method, discrete sine-Gordon equation, dynamical behaviors

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Authors: Masoud Ghermezi

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Studying the sheet metals is one of the most important research areas in the field of metal forming due to their extensive applications in the aerospace industries. A useful method for determining the forming limit of these materials and consequently preventing the rupture of sheet metals during the forming process is the use of the forming limit curve (FLC). In addition to specifying the forming limit, this curve also delineates a boundary for the allowed values of strain in sheet metal forming; these characteristics of the FLC along with its accuracy of computation and wide range of applications have made this curve the basis of research in the present paper. This study presents a new model that not only agrees with the results obtained from the above mentioned theory, but also eliminates its shortcomings. In this theory, like in the M-K theory, a thin sheet with an inhomogeneity as a gradient thickness reduction with a sinusoidal function has been chosen and subjected to two-dimensional stress. Through analytical evaluation, ultimately, a governing differential equation has been obtained. The numerical solution of this equation for the range of positive strains (stretched region) yields the results that agree with the results obtained from M-K theory. Also the solution of this equation for the range of negative strains (tension region) completes the FLC curve. The findings obtained by applying this equation on two alloys with the hardening exponents of 0.4 and 0.24 indicate the validity of the presented equation.

Keywords: sheet metal, metal forming, forming limit curve (FLC), M-K theory

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Authors: Alexander P. Smirnov, Sergey A. Matveev, Dmitry A. Zheltkov, Eugene E. Tyrtyshnikov

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We propose a new computational technique for multidimensional (multicomponent) Smoluchowski coagulation equation. Using low-rank approximations in Tensor Train format of both the solution and the coagulation kernel, we accelerate the classical finite-difference Runge-Kutta scheme keeping its level of accuracy. The complexity of the taken finite-difference scheme is reduced from O(N^2d) to O(d^2 N log N ), where N is the number of grid nodes and d is a dimensionality of the problem. The efficiency and the accuracy of the new method are demonstrated on concrete problem with known analytical solution.

Keywords: tensor train decomposition, multicomponent Smoluchowski equation, runge-kutta scheme, convolution

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Authors: A. Guezane-Lakoud, S. Bensebaa

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In this paper, we study a boundary value problem of nonlinear fractional differential equation. Existence and positivity results of solutions are obtained.

Keywords: positive solution, fractional caputo derivative, Banach contraction principle, Avery and Peterson fixed point theorem

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Authors: Farid Nouioua, Abdelouaheb Ardjouni

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In this article, we study the existence of positive periodic solutions of certain delay differential equations. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ Krasnoselskii's fixed point theorem to obtain sufficient conditions for the existence of a positive periodic solution of the differential equation. The obtained results improve and extend the results in the literature. Finally, an example is given to illustrate our results.

Keywords: delay differential equations, positive periodic solutions, integral equations, Krasnoselskii fixed point theorem

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Authors: Shubham Jaiswal

Abstract:

In the present article, a drive is taken to compute the solution of spatial fractional order advection-dispersion equation having source/sink term with given initial and boundary conditions. The equation is converted to a system of ordinary differential equations using second-kind shifted Chebyshev polynomials, which have finally been solved using finite difference method. The striking feature of the article is the fast transportation of solute concentration as and when the system approaches fractional order from standard order for specified values of the parameters of the system.

Keywords: spatial fractional order advection-dispersion equation, second-kind shifted Chebyshev polynomial, collocation method, conservative system, non-conservative system

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Authors: Kajal K. Patel, M. N. Mehta, T. R. Singh

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When water is injected in oil formatted area in secondary oil recovery process the instability occurs near common interface due to viscosity difference of injected water and native oil. The governing equation gives rise to the non-linear partial differential equation and its solution has been obtained by Homotopy analysis method with appropriate guess value of the solution together with some conditions and standard relations. The solution gives the average cross-sectional area occupied by the schematic fingers during the occurs of instability phenomenon. The numerical and graphical presentation has developed by using Maple software.

Keywords: capillary pressure, homotopy analysis method, instability phenomenon, viscosity

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Authors: S. A. M. Yatim, Z. B. Ibrahim, K. I. Othman, M. Suleiman

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The method of solving second order stiff ordinary differential equation (ODEs) that is based on backward differentiation formula (BDF) is considered in this paper. We derived the method by increasing the order of the existing method using an improved strategy in choosing the step size. Numerical results are presented to compare the efficiency of the proposed method to the MATLAB’s suite of ODEs solvers namely ode15s and ode23s. The method was found to be efficient to solve second order ordinary differential equation.

Keywords: backward differentiation formulae, block backward differentiation formulae, stiff ordinary differential equation, variable step size

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Authors: Raphael Zanella

Abstract:

This works deals with the finite element approximation of axisymmetric problems. The weak formulation of the heat equation under the axisymmetry assumption is established for continuous finite elements. The weak formulation is implemented in a C++ solver with implicit march-in-time. The code is verified by space and time convergence tests using a manufactured solution. The solving of an example problem with an axisymmetric formulation is compared to that with a full-3D formulation. Both formulations lead to the same result, but the code based on the axisymmetric formulation is much faster due to the lower number of degrees of freedom. This confirms the correctness of our approach and the interest in using an axisymmetric formulation when it is possible.

Keywords: axisymmetric problem, continuous finite elements, heat equation, weak formulation

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Authors: Changhong Guo, Shaomei Fang, Yong He

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In this paper, fractional Black-Scholes models for the European option pricing were established based on the fractional G-Brownian motion (fGBm), which generalizes the concepts of the classical Brownian motion, fractional Brownian motion and the G-Brownian motion, and that can be used to be a tool for considering the long range dependence and uncertain volatility for the financial markets simultaneously. A generalized fractional Black-Scholes equation (FBSE) was derived by using the Taylor’s series of fractional order and the theory of absence of arbitrage. Finally, some explicit option pricing formulas for the European call option and put option under the FBSE were also solved, which extended the classical option pricing formulas given by F. Black and M. Scholes.

Keywords: European option pricing, fractional Black-Scholes equations, fractional g-Brownian motion, Taylor's series of fractional order, uncertain volatility

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Authors: Leonid Borisov, Yuri Orlov

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We obtain explicit formulas of finitely multiple approximations of the equilibrium density matrix for the case of the harmonic oscillator using Chernoff's theorem and the notion of semigroup which is Chernoff equivalent to average semigroup. Also we found explicit formulas for the corresponding approximate Wigner functions and average values of the observable. We consider a superposition of τ -quantizations representing a wide class of linear quantizations. We show that the convergence of the approximations of the average values of the observable is not uniform with respect to the Gibbs parameter. This does not allow to represent approximate expression as the sum of the exact limits and small deviations evenly throughout the temperature range with a given order of approximation.

Keywords: Chernoff theorem, Feynman formulas, finitely multiple approximation, harmonic oscillator, Wigner function

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Authors: Razieh Khalafi

Abstract:

The brain is the information processing centre of the human body. Stimuli in the form of information are transferred to the brain and then brain makes the decision on how to respond to them. In this research, we propose a new partial differential equation which analyses the EEG signals and make a relationship between the incoming stimuli and the brain response to them. In order to test the proposed model, a set of external stimuli applied to the model and the model’s outputs were checked versus the real EEG data. The results show that this model can model the EEG signal well. The proposed model is useful not only for modelling of EEG signal in case external stimuli but it can be used for modelling of brain response in case of internal stimuli.

Keywords: brain, stimuli, partial differential equation, response, EEG signal

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Authors: J. J. Peña, J. Morales, J. García-Ravelo, L. Arcos-Díaz

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The Point canonical transformation method is applied for solving the Schrödinger equation with position-dependent mass. This class of problem has been solved for continuous mass distributions. In this work, a staggered mass distribution for the case of a free particle in an infinite square well potential has been proposed. The continuity conditions as well as normalization for the wave function are also considered. The proposal can be used for dealing with other kind of staggered mass distributions in the Schrödinger equation with different quantum potentials.

Keywords: free particle, point canonical transformation method, position-dependent mass, staggered mass distribution

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