In this paper we first prove a theorem on the nonexistence of pyramidal polynomial basis functions. Then we present a new symmetric composite pyramidal finite element which yields a better convergence than the nonsymmetric one. It has fourteen degrees of freedom and its basis functions are incomplete piecewise triquadratic polynomials. The space of ansatz functions contains all quadratic functions on each of four sub-tetrahedra that form a given pyramidal element.

In this work we investigate the novel Kryzhnyi method for the numerical inverse Laplace transformation and apply it to the pricing problem of continuous installment options. We compare the results with the one obtained using other classical methods for the inverse Laplace transformation, like the Euler summation method or the Gaver-Stehfest method.

The energy gradient method has been proposed with the aim of better understanding the mechanism of flow transition from laminar flow to turbulent flow. In this method, it is demonstrated that the transition to turbulence depends on the relative magnitudes of the transverse gradient of the total mechanical energy which amplifies the disturbance and the energy loss from viscous friction which damps the disturbance, for given imposed disturbance. For a given flow geometry and fluid properties, when the maximum of the function $K$ (a function standing for the ratio of the gradient of total mechanical energy in the transverse direction to the rate of energy loss due to viscous friction in the streamwise direction) in the flow field is larger than a certain critical value, it is expected that instability would occur for some initial disturbances. In this paper, using the energy gradient analysis, the equation for calculating the energy gradient function $K$ for plane Couette flow is derived. The result indicates that $K$ reaches the maximum at the moving walls. Thus, the fluid layer near the moving wall is the most dangerous position to generate initial oscillation at sufficient high $\operatorname{Re}$ for given same level of normalized perturbation in the domain. The critical value of $K$ at turbulent transition, which is observed from experiments, is about 370 for plane Couette flow when two walls move in opposite directions (anti-symmetry). This value is about the same as that for plane Poiseuille flow and pipe Poiseuille flow (385-389). Therefore, it is concluded that the critical value of $K$ at turbulent transition is about 370-389 for wall-bounded parallel shear flows which include both pressure (symmetrical case) and shear driven flows (anti-symmetrical case).

In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the $L^2$-norm error over a finite time interval converges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method. 

This paper concentrates on the wave motion at the interface of viscous compressible fluid half-space and homogeneous isotropic, generalized thermoelastic diffusive half-space. The wave solutions in both the fluid and thermoelastic diffusive half-spaces have been investigated; and the complex dispersion equation of leaky Rayleigh wave motion have been derived. The phase velocity and attenuation coefficient of leaky Rayleigh waves have been computed from the complex dispersion equation by using the Muller's method. The amplitudes of displacements, temperature change and concentration have been obtained. The effects of viscosity and diffusion on phase velocity and attenuation coefficient of leaky Rayleigh waves motion for different theories of thermoelastic diffusion have been depicted graphically. The magnitude of heat and mass diffusion flux vectors for different theories of thermoelastic diffusion have also been computed and represented graphically.

This paper is concerned with a stabilized finite element method based on two local Gauss integrations for the two-dimensional non-stationary conduction-convection equations by using the lowest equal-order pairs of finite elements. This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf-sup condition. The stability of the discrete scheme is derived under some regularity assumptions. Optimal error estimates are obtained by applying the standard Galerkin techniques. Finally, the numerical illustrations agree completely with the theoretical expectations.