Exact solutions of multi-term fractional difusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional difusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial diferential equations are converted into time-fractional ordinary diferential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-difusion problems are given to validate the proposed analytical method.     

Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain

A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors.     

General solutions to a class of time fractional partial differential equations

A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier- Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine- Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.     

Exact solutions in generalized Oldroyd-B fluid

This investigation deals with the influence of slip condition on the magnetohydrodynamic (MHD) and rotating flow of a generalized Oldroyd-B (G.Oldroyd-B)fluid occupying a porous space.Fractional calcul...     

Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

The velocity field and the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. Initially the fluid is at rest, and at time t = 0⁺, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary second grade and Newtonian fluids are obtained as limiting cases of the general solutions. Finally, some characteristics of the motion, as well as the influences of the material and fractional parameters on the fluid motion and a comparison between models, are underlined by graphical illustrations.     

Explicit solitary-wave solutions to generalized Pochhammer-Chree equations

ｎｔｒｏｄｕｃｔｉｏｎＰｏｃｈｈａｍｍｅｒ＿Ｃｈｒｅｅｑｕａｔｉｏｎ（ＰＣｅｑｕａｔｉｏｎｉｎｓｈｏｒｔ）ｕｔ－ｕｔｘｘ－ｕｘｘ－１ｐ（ｕｐ）ｘｘ＝０,（１）ｉｓｕｓｅｄｔｏｄｅｓｃｒｉｂｅｔｈｅｐｒｏｐａｇａｔｉｏｎｏｆｌｏｎｇｉｔｕｄｉｎａｌｄｅ...     

分数阶微分方程的Haar小波算法研究

在BPFs的Caputo分数阶微分算子矩阵的基础上,建立了Haar小波的分数阶微分算子矩阵,提出了一种有效的求解分数阶微分方程的Haar小波数值方法,并将该方法应用于线性和非线性分数阶常微分方程求解中.数值算例表明,该算法简单,数值精确度高,是一种高效的数值求解方法.     

Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions

We give an analytical treatment of a time fractional diffusion equation with Caputo time-fractional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experiment in limiting diffusion space.     

Backlund transformation and exact solutions for whitham-broer-kaup equations in shallow water

I.IntroductionTodescribethepropagationofshallowwaterwave,manywell-knowncompletelyintegrablemodelsareintroduced,suchasKdVequatioll,Boussinesqequation,K-Pequation,WBKequation,etc.UnderBoussinesqapproximation,Whitham,BroerandKaupl"2'3]obtainednon-lilleurWBKequationwhereu=u(x,l)isthefieldofhorizontalvelocity;v=v(x,t)istheheightthatdeviatefromequilibriumpositionofliquid;a,gareconstantsthatrepresentdift'erentdispersivepower.TheEqs.(1.I),(l.2)areverygoodmodelstodescl.ibedispersivewave.Ifa=0,P…     

A semi-analysis method of differential equations with variable coefficients under complicated boundary conditions

IntroductionInthemostcases,asfarasthesolutiontothedifferentialequationwithvariablecoefficientsisconcerned ,therehasbeennoanysatisfactoryanswer.Forthisreason ,thescholarsbothathomeandabroadhavedonesomeresearchjustasindicatedinRefs.[1 ] ,[2 ]and [3 ] ,etc.Theauthorsinthispaperhaveusedthefiniteelementmodelandthemetricfunctiontheoryincombinationwithadjustableparametermodelwithvariablecoefficientstosimplifythesolutiontothedifferentialequationwithvariablecoefficientsintothefeaturevalueaswellasthefea…     

Exact solutions for nonlinear transient flow model including a quadratic gradient term

The models of the nonlinear radial flow for the infinite and finite reservoirs including a quadratic gradient term were presented. The exact solution was given in real space for flow equation including quadratic gradiet term for both constant-rate and constant pressure production cases in an infinite system by using generalized Weber transform.Analytical solutions for flow equation including quadratic gradient term were also obtained by using the Hankel transform for a finite circular reservoir case. Both closed and constant pressure outer boundary conditions are considered. Moreover, both constant rate and constant pressure inner boundary conditions are considered. The difference between the nonlinear pressure solution and linear pressure solution is analyzed. The difference may be reached about 8% in the long time. The effect of the quadratic gradient term in the large time well test is considered.     

A numerical method for fractional integral with applications

IntroductionThefractionalcalculushasalonghistoryandthereareamassofworkstodiscussthefractionalderivativesandfractionalintegralswitharbitrary (realorcomplex)order[1- 3 ].Thefractionalcalculushasawideapplicationbackground ,especiallyinthefieldsofchemistry ,electromagnetics,materialscienceandmechanics.Forexample,Gement[4 ]proposedthefractionalderivativeconstitutivemodelsofaviscoelasticmaterialatfirst.Themodelshavereceivedincreasingattention[5 - 7].Onlyafewparametersarecontainedinthemodelsandthemo…     

A higher order numerical scheme for generalized fractional diffusion equations

In this article, we develop a higher order approximation for the generalized fractional derivative that includes a scale function z(t) and a weight function w(t). This is then used to solve a generalized fractional diffusion problem numerically. The stability and convergence analysis of the numerical scheme are conducted by the energy method. It is proven that the temporal convergence order is 3 and this is the best result to date. Finally, we present four examples to confirm the theoretical results.     

Radiation boundary conditions for finite element solutions of generalized wave equations

On the basis of the dispersion relation of the generalized linear wave equation we derive a radiation boundary condition (RBC) that explicitly incorporates the physical parameters of the governing equation into the form of the boundary condition. Using finite element techniques we investigate the properties of the generalized RBC by examining forced and unforced solutions to the telegraph and Klein-Gordon equations in one dimension. The results show that within the limits of the physical parameters of the problem the generalized RBC is an improvement over the Sommerfeld RBC when the governing equation contains additional terms that influence the propagation. These gains are achieved without introducing any computational overhead. A two-dimensional example suggests that the 1D findings can generalize to higher dimensions.     

Viscoplastic field near a propagating crack-tip

An elastic-viscoplastic constitutive model is proposed instead of the usual elastoplastic model. It is assumed that when crack-tip is approached the viscosity coefficient lends to zero (=₀r). Asymptotic analysis of the dynamic field near a propagating crack-tip is given, and the uniparameter solution is obtained. The numerical result is given for various Mach number and viscosity coefficient. Based on the asymptotic solution, a fracture criterion is proposed and the stability of crack propagation is discussed.     

Three-dimensional exact solutions for the free vibration of laminated transversely isotropic circular,annular and sectorial plates with unusual boundary conditions

New state space formulations for the free vibration of circular, annular and sectorial plates are established by introducing two displacement functions and two stress functions. The state variables can be separated into two independent catalogues and two kinds of vibrations can be readily found. Expanding the displacements and stresses in terms of Bessel functions in the radial direction and trigonometric functions in the circumferential direction, we obtained the exact frequency equation for the free vibration for some uncommon boundary conditions. Numerical results are presented and compared with those of FEM to demonstrate the reliability of the proposed method. A parametric investigation is also performed.     

Uniqueness of solutions for a class of non-linear volterra integral equations without continuity

I.IntroductionItiswell-knownthatthenon-linearVolterraintegralequationn(x)~10k(x~s)g(u(s))ds(0相似文献   

Periodic solutions of a class of higher order neutral type equations

In this paper, by ssing the theory of Fourier series, some necessary and sufficient conditions of existence and uniqueness of periodic solutions of a class of higher order neutral type equations are obtained. The main results by Shi Jianguo in “Discussion on the periodic solutions for linear equation of neutral type with constant coefficients” are improved, i. e., the condition |b₀|≠1 instead of the condtion |b₀|<1/2 of Theorem 1 by Shi Jianguo is given. Other theorems by Shi are rebuilt and improved according to the new assumption. Foundation item: the Natural Science Foundation of Yunnan Province, China (97A012G)     

Exact solutions for free vibration of transversely isotropic piezoelectric circular plates

By employing the general solution for coupled three-dimensional equations of a transversely isotropic piezoelectric body, this paper investigates the free vibration of a circular plate made of piezoelectric material. Three-dimensional exact solutions are then obtained under two specified boundary conditions, which can be used for both axisymmetric and non-axisymmetric cases. Numerical results are finally presented. The project supported by the National Natural Science Foundation of China (No. 19872060)     

Exact analysis of the orientation-adjusted adhesive full stick contact of layered structures with the asymmetric bipolar coordinates

The adhesion failure has become one dominant factor in determining the reliability and service life of miniaturized devices subject to loadings with arbitrary orientations. This article establishes an adhesive full stick contact model between an elastic half-space and a rigid cylinder loaded in any direction. Using the Papkovich-Neuber functions, the Fourier integral transform, and the asymmetric bipolar coordinates, the exact solution is obtained. Unlike the Johnson-Kendall-Roberts (JKR) model, the present adhesive contact model takes into account the effects of the load direction as well as the coupling of the normal and tangential contact stresses. Besides, it considers the full stick contact which has large values of the friction coefficient between contacting surfaces, contrary to the frictionless contact supposed in the JKR model. The result shows that suitable angles can be found, which makes the contact surfaces difficult to be peeled off or easy to be pressed into.