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.     

Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media

A theoretical analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media is studied. Using the method of symmetry group of scaling transformations and the H-function, the analytical solutions of concentration distribution are given. At the same time we derive the expressions of scattering function spectrum.The result shows that the scattering function spectra still have the properties of scaling function. The scattering functions of point source, line source and area source in regular Euclidean space can be regarded as particular cases of this paper and are included in this paper. At the end of the paper we discuss the asymptotic behaviors of the solution in detail. The results of this paper can be taken to be the fundamental solutions for every kind of boundary value problems of fractional anomalous diffusion in disordered fractal media.     

Discretization of an admixture flux within the framework of the fractal model of anomalous diffusion

Within the framework of the fractal mobile-immobile medium model describing non-Fickian effects occurring in admixture seepage due to particle adhesion to the solid matrix, an expression for the admixture flux is derived. Flow discretization intended for finite-difference calculations is proposed and used as a basis for a conservation-law scheme for solving the model equations with account for admixture sources. Several one-dimensional test problems of admixture propagation in an imposed seepage flow are solved using the approach developed.     

Exact solution for elliptical inclusion in magnetoelectroelastic materials

This paper considers the magneto-electro-mechanical coupling between an inclusion and matrix, which are both of magnetoelectroelastic materials. The general cases including the mode I, mode II and mode III are studied. Analytical solutions for an elliptical cylinder inclusion inside an infinite magnetoelectroelastic medium under combined mechanical–electrical–magnetic loadings are formulated via the Stroh formalism. Crack problem is also investigated and the stress, electric and magnetic fields in the vicinity of the crack tip are determined by a complex vector of intensity factors. Various special cases, including an impermeable inclusion, a permeable crack, a rigid and permeable inclusion, a rigid and permeable line inclusion, a permeable cavity, and an impermeable cavity are discussed.     

Anomalous diffusion and anomalous stretching in vortical flows *

The trajectories of particles advected by non-random spatially periodic velocity fields may be chaotic, i.e., extremely sensitive to their initial positions. In the most general case, the area of a cloud of advected particles grows like (time) where the exponent ranges between 0 and 2 (shear stretching). These observations could explain many known facts about dispersion processes in shallow tidal seas.     

Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations

This paper presents an algorithm to obtain numerically stable differentiation matrices for approximating the left- and right-sided Caputo-fractional derivatives. The proposed differentiation matrices named fractional Chebyshev differentiation matrices are obtained using stable recurrence relations at the Chebyshev–Gauss–Lobatto points. These stable recurrence relations overcome previous limitations of the conventional methods such as the size of fractional differentiation matrices due to the exponential growth of round-off errors. Fractional Chebyshev collocation method as a framework for solving fractional differential equations with multi-order Caputo derivatives is also presented. The numerical stability of spectral methods for linear fractional-order differential equations (FDEs) is studied by using the proposed framework. Furthermore, the proposed fractional Chebyshev differentiation matrices obtain the fractional-order derivative of a function with spectral convergence. Therefore, they can be used in various spectral collocation methods to solve a system of linear or nonlinear multi-ordered FDEs. To illustrate the true advantages of the proposed fractional Chebyshev differentiation matrices, the numerical solutions of a linear FDE with a highly oscillatory solution, a stiff nonlinear FDE, and a fractional chaotic system are given. In the first, second, and forth examples, a comparison is made with the solution obtained by the proposed method and the one obtained by the Adams–Bashforth–Moulton method. It is shown the proposed fractional differentiation matrices are highly efficient in solving all the aforementioned examples.     

Exact solution and stability of postbuckling configurations of beams

We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.     

Analytical solution for one-dimensional coupled non-Fick diffusion and mechanics

On the basis of the developed coupled non-Fick diffusion and mechanics theory, the approximate analytical solutions of concentration, stress and displacement are respectively derived by using the Laplace transform and inverse Laplace transform. There exist two discontinuous points in both the stress and concentration solutions, while the displacement is continuous. Moreover, the two discontinuous points verify the interaction between the non-Fick diffusion and mechanics. Then, numerical examples about the stress, concentration and displacement field are carried out to show the coupled behavior. Finally, the non-Fick effect is graphically obvious in the results.     

Exact solution of electroosmotic flow in generalized Burgers fluid

An exact solution is developed for the time periodic electroosmotic flow of a non-Newtonian fluid between the micro-parallel plates. The constitutive equations of a generalized Burgers fluid are used in the mathematical formulation. The resulting problem is solved by a Fourier transform technique. Graphs are plotted and discussed for various emerging parameters of interest.     

Exact solution for the unsteady flow of a semi-infinite micropolar fluid

The unsteady motion of an incompressible micropolar fluid filling a half-space bounded by a horizontal infinite plate that started to move suddenly is considered. Laplace transform techniques are used. The solution in the Laplace transform domain is obtained by using a direct approach. The inverse Laplace transforms are obtained in an exact manner using the complex inversion formula of the transform together with contour integration techniques. The solution in the case of classical viscous fluids is recovered as a special case of this work when the micropolarity coecient is assumed to be zero. Numerical computations are carried out and represented graphically.     

Exact solution for free vibration analysis of an eccentric elliptical plate

The method of separation of variables in elliptical coordinates in conjunction with the translational addition theorems for Mathieu functions is used to investigate the free flexural vibrations of a fully clamped thin elastic panel of elliptical planform containing an elliptical cutout of arbitrary size, location, and orientation. The first five natural frequencies are calculated for various plate/cutout aspect ratios and selected cutout location/orientation parameters. Also, a number of representative vibration mode shapes are depicted in graphical form. The accuracy of solutions is demonstrated through proper convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with those in the existing literature.     

Exact statement of the instability problem for frames and its direct numerical solution

A general approach for the systematic evaluation of the critical buckling load and the determination of the buckling mode is presented. The Navier-Bernoulli beam model is considered, having the possibility of variable cross-section under any type of load (including pressures and thermal loading). With this purpose, the equilibrium equations of each beam element in its deformed configuration under the hypothesis of infinitesimal strains and displacements is considered, resulting in a system of differential equations with variable coefficients for each element. To obtain the nonlinear response of the frame, one should impose the compatibility of displacements and the equilibrium of forces and moments in each beam-end, also in the deformed configuration. The solution is obtained by requiring that the total variation of potential energy is zero at the instant of buckling. The objective of this work is to develop a systematic method to determine the critical buckling load and the bucklingmode of any frame without using the common simplifications usually assumed in matrix analysis or finite element approaches. This way, precise results can be obtained regardless of the discretization done.     

Exact solution of conductive heat transfer in cylindrical composite laminate

This paper presents an exact solution for steady-state conduction heat transfer in cylindrical composite laminates. This laminate is cylindrical shape and in each lamina, fibers have been wound around the cylinder. In this article heat transfer in composite laminates is being investigated, by using separation of variables method and an analytical relation for temperature distribution in these laminates has been obtained under specific boundary conditions. Also Fourier coefficients in each layer obtain by solving set of equations that related to thermal boundary layer conditions at inside and outside of the cylinder also thermal continuity and heat flux continuity between each layer is considered. In this research LU factorization method has been used to solve the set of equations.     

A dynamic framework for functional parameterisations of the eddy viscosity coefficient in two-dimensional turbulence

This paper puts forth a dynamic framework for investigating the subgrid scale physics of decaying two-dimensional turbulence utilising a modular approach with eddy viscosities in various functional forms. The derivation of the low-pass spatially filtered implementation of the Navier–Stokes equations is given by using the vorticity-streamfunction formulation. Two different implementations of the viscosity kernels based on the representation of the eddy viscosity terms are proposed and tested by solving a canonical two-dimensional decaying turbulence problem. It is seen that the implementation of the eddy viscosity formulation plays a distinct role in the dissipative behaviour of the different viscosity kernels. Among eddy viscosity kernels tested, we found that the Leith eddy viscosity formulation yields superior results with higher correlation coefficients.     

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.