Approximate solution of two-term fractional-order diffusion,wave-diffusion,and telegraph models arising in mathematical physics using optimal homotopy asymptotic method

This paper deals with the investigation of the analytical approximate solutions for two-term fractional-order diffusion, wave-diffusion, and telegraph equations. The fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], (1,2), and [1,2], respectively. In this paper, we extended optimal homotopy asymptotic method (OHAM) for two-term fractional-order wave-diffusion equations. Highly approximate solution is obtained in series form using this extended method. Approximate solution obtained by OHAM is compared with the exact solution. It is observed that OHAM is a prevailing and convergent method for the solutions of nonlinear-fractional-order time-dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective, and easy to use, for handling more general fractional-order wave diffusion, diffusion, and telegraph problems.     

Vibrations of double-string complex system under moving forces. Closed solutions

In this paper the dynamic response of a double-string system traversed by a constant or a harmonically oscillating moving force is considered. The force is moving with a constant velocity on the top string. The strings are identical, parallel, one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a double-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case a part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations.     

Transverse vibrations of hybrid laminated plates

In this paper, an attempt is made to obtain the free vibration response of hybrid, laminated rectangular and skew plates. The Galerkin technique is employed to obtain an approximate solution of the governing differential equations. It is found that this technique is well suited for the study of such problems. Results are presented in a graphical form for plates with one pair of opposite edges simply supported and the other two edges clamped. The method is quite general and can be applied to any other boundary conditions.     

The operational solution of fractional-order differential equations,as well as Black–Scholes and heat-conduction equations

Operational solutions to fractional-order ordinary differential equations and to partial differential equations of the Black–Scholes and of Fourier heat conduction type are presented. Inverse differential operators, integral transforms, and generalized forms of Hermite and Laguerre polynomials with several variables and indices are used for their solution. Examples of the solution of ordinary differential equations and extended forms of the Fourier, Schrödinger, Black–Scholes, etc. type partial differential equations using the operational method are given. Equations that contain the Laguerre derivative are considered. The application of the operational method for the solution of a number of physical problems connected with charge dynamics in the framework of quantum mechanics and heat propagation is demonstrated.     

Initialized fractional differential equations with Riemann-Liouville fractional-order derivative

The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.     

An efficient technique for solving the space-time fractional reaction-diffusion equation in porous media

In this paper, we obtained the approximate numerical solution of space-time fractional-order reaction-diffusion equation using an efficient technique homotopy perturbation technique using Laplace transform method with fractional-order derivatives in Caputo sense. The solution obtained is very useful and significant to analyze the many physical phenomenons. The present technique demonstrates the coupling of the homotopy perturbation technique and Laplace transform using He’s polynomials for finding the numerical solution of various non-linear fractional complex models. The salient features of the present work are the graphical presentations of the approximate solution of the considered porous media equation for different particular cases and reflecting the presence of reaction terms presented in the equation on the physical behavior of the solute profile for various particular cases.     

Novel approach for modified forms of Camassa-Holm and Degasperis-Procesi equations using fractional operator

In the present study, we consider the q-homotopy analysis transform method to find the solution for modified Camassa-Holm and Degasperis-Procesi equations using the Caputo fractional operator. Both the considered equations are nonlinear and exemplify shallow water behaviour. We present the solution procedure for the fractional operator and the projected solution procedure gives a rapidly convergent series solution. The solution behaviour is demonstrated as compared with the exact solution and the response is plotted in 2D plots for a diverse fractional-order achieved by the Caputo derivative to show the importance of incorporating the generalised concept. The accuracy of the considered method is illustrated with available results in the numerical simulation. The convergence providence of the achieved solution is established in $\hslash $-curves for a distinct arbitrary order. Moreover, some simulations and the important nature of the considered model, with the help of obtained results, shows the efficiency of the considered fractional operator and algorithm, while examining the nonlinear equations describing real-world problems.     

Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

In this paper, the formulation of time-fractional (TF) electrodynamics is derived based on the Riemann-Silberstein (RS) vector. With the use of this vector and fractional-order derivatives, one can write TF Maxwell’s equations in a compact form, which allows for modelling of energy dissipation and dynamics of electromagnetic systems with memory. Therefore, we formulate TF Maxwell’s equations using the RS vector and analyse their properties from the point of view of classical electrodynamics, i.e., energy and momentum conservation, reciprocity, causality. Afterwards, we derive classical solutions for wave-propagation problems, assuming helical, spherical, and cylindrical symmetries of solutions. The results are supported by numerical simulations and their analysis. Discussion of relations between the TF Schrödinger equation and TF electrodynamics is included as well.     

扩散沟道光波导的条形传递函数解

利用条形传递函数方法分析了二维扩散沟道光波导的传播特性,对两种典型折射率分布计算的数值结果与精确解及其它方法的结果进行了比较,表明该方法具有优于其它数值方法的精度.     

Fractional Klein-Gordon-Schrödinger equations with Mittag-Leffler memory

The main objective of the present investigation is to find the solution for the fractional model of Klein-Gordon-Schrödinger system with the aid of q-homotopy analysis transform method (q-HATM). The projected solution procedure is an amalgamation of q-HAM with Laplace transform. More preciously, to elucidate the effectiveness of the projected scheme we illustrate the response of the q-HATM results, and the numerical simulation is offered to guarantee the exactness. Further, the physical behaviour has been presented associated with parameters present the method with respect fractional-order. The present study confirms that, the projected solution procedure is highly methodical and accurate to solve and study the behaviours of the system of differential equations with arbitrary order exemplifying the real word problems.     

Macroscopic description of a subdiffusion-controlled bimolecular reaction

The random trap model is used to derive equations describing reaction-subdiffusion systems with diffusion-controlled (infinitely fast) bimolecular reaction. A hierarchy of equations in terms of distribution functions is closed by using a quasi-equilibrium condition in the equation for the two-particle distribution function. The reaction terms in the resulting equations contain products between concentrations and diffusion jump rates, rather than products of concentrations as dictated by the law of mass action. The same equations are also derived in the framework of a nonlinear continuous-time random walk model. The equations are used to show that inhomogeneity of the medium may manifest itself by fractional-order reaction terms.     

APPROXIMATE ANALYTICAL SOLUTIONS FOR PRIMARY CHATTER IN THE NON-LINEAR METAL CUTTING MODEL

This paper considers an accepted model of the metal cutting process dynamics in the context of an approximate analysis of the resulting non-linear differential equations of motion. The process model is based upon the established mechanics of orthogonal cutting and results in a pair of non-linear ordinary differential equations which are then restated in a form suitable for approximate analytical solution. The chosen solution technique is the perturbation method of multiple time scales and approximate closed-form solutions are generated for the most important non-resonant case. Numerical data are then substituted into the analytical solutions and key results are obtained and presented. Some comparisons between the exact numerical calculations for the forces involved and their reduced and simplified analytical counterparts are given. It is shown that there is almost no discernible difference between the two thus confirming the validity of the excitation functions adopted in the analysis for the data sets used, these being chosen to represent a real orthogonal cutting process. In an attempt to provide guidance for the selection of technological parameters for the avoidance of primary chatter, this paper determines for the first time the stability regions in terms of the depth of cut and the cutting speed co-ordinates.     

Nonuniform autonomous one-dimensional exclusion nearest-neighbor reaction-diffusion models

The most general nonuniform reaction-diffusion models on a one-dimensional lattice with boundaries, for which the time evolution equations of correlation functions are closed, are considered. A transfer matrix method is used to find the static solution. It is seen that this transfer matrix can be obtained in a closed form, if the reaction rates satisfy certain conditions. We call such models superautonomous. Possible static phase transitions of such models are investigated. At the end, as an example of superautonomous models, a nonuniform voter model is introduced, and solved explicitly.     

Soliton solutions for the space-time nonlinear partial differential equations with fractional-orders

Many practical models in interdisciplinary fields can be described with the help of fractional-order nonlinear partial differential equations(NPDEs). Fractional-order NPDEs such as the space-time fractional Fokas equation, the space-time Kaup–Kupershmidt equation and the space-time fractional (2+1)-dimensional breaking soliton equation have been widely applied in many branches of science and engineering. So, finding exact traveling wave solutions are very helpful in the theories and numerical studies of such equations. More precisely, fractional sub-equation method together with the proposed technique is implemented to obtain exact traveling wave solutions of such physical models involving Jumarie’s modified Riemann–Liouville derivative. As a result, some new exact traveling wave solutions for them are successfully established. Also, (1+1)-dimensional plots and 1-dimensional plots of some of the derived solutions are given to visualize the dynamics of the considered NPDEs. The obtained results reveal that the proposed technique is quite effective and convenient for obtaining exact solutions of NPDEs with fractional-order.     

An exact frequency equation in closed form for Timoshenko beam clampled at both ends

The author has discovered several errors which are not typographical in the frequency equations for a Timoshenko beam clamped at both ends by Huang who presented the frequency equations and normal mode equations for all six common types of simple, finite beams in closed form for the first time. The exact frequency equations in closed form for Timoshenko beams clamped at both ends are derived based on his analysis. And then in order to justify the amended solutions of Huang, two versions of the closed form exact method and the Ritz method are applied. The frequency equations by the previous researcher present frequencies for only the flexural modes, while the closed form exact method and the Ritz method give ones for the thickness–shear modes as well as the bending modes. The purpose of the present study is to reveal the errors, correct them, and give some numerical results.     

A Method for Solving Initial-value Problem and Finding Exact Solutions of Nonlinear Partial Differential Equations

Homogeneous balance method for solving nonlinear partial differential equation(s) is extended to solving initial-value problem and getting new solution(s) from a known solution of the equation(s) under consideration. The approximate equations for long water waves are chosen to illustrate the method, infinitely many simple-solitary-wave solutions and infinitely many rational function solutions, especially the closed form of the solution for initial-value problem, are obtained by using the extended homogeneous balance method given here.     

Synthesis of mode converters on the basis of the FDTD method

We propose a universal technique for synthesizing mode converters, which is based on numerical integration of the Maxwell equations on a space-time mesh by the FDTD method. The new technique is an iterative algorithm, in which the correction to the converter profile at each iteration is calculated via the fields on the converter surface that are obtained from two conjugate problems, specifically, by direct and inverse (with an inversion of the time-integration direction) solution of the equations for electromagnetic fields. The efficiency of the synthesis algorithm is illustrated by examples that are of practical importance. The technique is compared with that proposed earlier, which used the solution of a system of equations for coupled waves. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 52, No. 3, pp. 216–230, March 2009.     

On the existence of a general rotating solution of Einstein's equations

In an earlier paper we considered a power-series expansion of the metric for a rotating field in terms of a parameter and constructed a solution of Einstein's equations to the first few orders in terms of two harmonic functions. We encountered a pair of Poisson-type equations which were apparently insoluble explicitly. The form of the metric considered was the Weyl-Lewis-Papapetrou form. In this paper we consider a power-series expansion of the most general form of a rotating metric and show that one encounters the same two Poisson equations as before. If these equations are insoluble explicitly, as seems likely, then a general solution depending on two harmonic functions cannot exist in closed form.     

A computational fluid dynamic technique valid at the centerline for non-axisymmetric problems in cylindrical coordinates

A technique is described for the numerical solution of non-axisymmetric flow problems posed in cylindrical coordinates when the z-axis is included in the flowfield. The highlight of the technique is the manner in which the singularities at the centerline are handled. Specifically, the governing flowfield equations at R = 0 are put in a special form by applying L'Hospital's Rule. The required radial derivatives are evaluated using a one-sided, second-order accurate, first-difference. This leads to a smooth, convergent calculation of the flow-field at the centerline. This appears to be the first generally applicable numerical method for avoiding coordinate system singularites in the context of a finite-difference scheme, and could have application to many non-axisymmetric flows. The technique is illustrated by specific results for the time-dependent flowfield inside an internal combustion engine.     

A new numerical solution for the MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method

In this Letter, we considered a numerical treatment for the solution of the hydromagnetic peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube using Adomian decomposition method and a modified form of this method. The axial velocity is obtained in a closed form. Comparison is made between the results obtained by only three terms of Adomian series with those obtained previously by perturbation technique. It is observed that only few terms of the series expansion are required to obtain the numerical solution with good accuracy.