Steady incompressible flow around a circular cylinder in an external magnetic field that is aligned with fluid flow direction is studied for $\text{Re}$ (Reynolds number) up to 40 and the interaction parameter in the range $0\le N\le 15$ (or $0\le M\le 30$), where $M$ is the Hartmann number related to $N$ by the relation $M=\sqrt{2N\text{Re}}$, using finite difference method. The pressure-Poisson equation is solved to find pressure fields in the flow region. The multigrid method with defect correction technique is used to achieve the second-order accurate solution of complete nonlinear Navier-Stokes equations. It is found that the boundary layer separation at rear stagnation point for $\text{Re}=10$ is suppressed completely when $N<1$ and it started growing again when $N\ge 9$. For $\text{Re}=20$ and 40, the suppression is not complete and in addition to that the rear separation bubble started increasing when $N\ge 3$. The drag coefficient decreases for low values of $N$ $\left(<0.1\right)$ and then increases with increase of $N$. The pressure drag coefficient, total drag coefficient, and pressure at rear stagnation point vary with $\sqrt{N}$. It is also found that the upstream and downstream pressures on the surface of the cylinder increase for low values of $N$ $\left(<0.1\right)$ and rear pressure inversion occurs with further increase of $N$. These results are in agreement with experimental findings.

We construct Jacobi-weighted orthogonal polynomials ${𝒫}_{n,r}^{\left(\alpha ,\beta ,\gamma \right)}\left(u,v,w\right),\alpha ,\beta ,\gamma >-1,\alpha +\beta +\gamma =0$, on the triangular domain $T$. We show that these polynomials ${𝒫}_{n,r}^{\left(\alpha ,\beta ,\gamma \right)}\left(u,v,w\right)$ over the triangular domain $T$ satisfy the following properties: ${𝒫}_{n,r}^{\left(\alpha ,\beta ,\gamma \right)}\left(u,v,w\right)\in {ℒ}_n,n\ge 1$, $r=0,1,\dots ,n,$ and ${𝒫}_{n,r}^{\left(\alpha ,\beta ,\gamma \right)}\left(u,v,w\right)\perp {𝒫}_{n,s}^{\left(\alpha ,\beta ,\gamma \right)}\left(u,v,w\right)$ for $r\ne s$. And hence, ${𝒫}_{n,r}^{\left(\alpha ,\beta ,\gamma \right)}\left(u,v,w\right)$, $n=0,1,2,\dots$, $r=0,1,\dots ,n$ form an orthogonal system over the triangular domain $T$ with respect to the Jacobi weight function. These Jacobi-weighted orthogonal polynomials on triangular domains are given in Bernstein basis form and thus preserve many properties of the Bernstein polynomial basis.

We prove existence and uniform stability of strong solutions to a quasilinear wave equation with a locally distributed nonlinear dissipation with source term of power nonlinearity of the type $u^{″}-M\left({\int }_{\Omega }{\left|\nabla u\right|}^{2}dx\right)\Delta u+a\left(x\right)g\left(u^{\prime }\right)+f\left(u\right)=0,$ in $\Omega \times \left]0,+\infty \right[$, $u=0,$ on $\Gamma \times \left]0,+\infty \right[,$ $u\left(x,0\right)=u_0\left(x\right),u^{\prime }\left(x,0\right)=u_1\left(x\right)$, in $\Omega$.

We analyse a model for pricing derivative securities in the presence of both transaction costs as well as the risk from a volatile portfolio. The model is based on the Black-Scholes parabolic PDE in which transaction costs are described following the Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by the variance of the synthesized portfolio. Transaction costs as well as the volatile portfolio risk depend on the time lag between two consecutive transactions. Minimizing their sum yields the optimal length of the hedge interval. In this model, prices of vanilla options can be computed from a solution to a fully nonlinear parabolic equation in which a diffusion coefficient representing volatility nonlinearly depends on the solution itself giving rise to explaining the volatility smile analytically. We derive a robust numerical scheme for solving the governing equation and perform extensive numerical testing of the model and compare the results to real option market data. Implied risk and volatility are introduced and computed for large option datasets. We discuss how they can be used in qualitative and quantitative analysis of option market data.

A theoretical analysis of the Saint-Venant one-dimensional flow model is performed in order to define the nature of its instability. Following the Brigg criterion, the investigation is carried out by examining the branch points singularities of dispersion relation in the complex $\omega$ and $k$ planes, where $\omega$ and $k$ are the complex pulsation and wave number of the disturbance, respectively. The nature of the linearly unstable conditions of flow is shown to be of convective type, independently of the Froude number value. Starting from this result a linear spatial stability analysis of the one-dimensional flow model is performed, in terms of time asymptotic response to a pointwise time periodic disturbance. The study reveals an influence of the disturbance frequency on the perturbation spatial growth rate, which constitutes the theoretical foundation of semiempirical criteria commonly employed for predicting roll waves occurrence.

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert space $l^2\times l^2$. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

The plane strain problem of determining strain energy release rate, crack energy, and crack-opening displacement (COD) for a moving Griffith crack at the interface of two dissimilar orthotropic half-planes is considered. The problem is reduced to a pair of singular integral equations of second kind which have finally been solved by using Jacobi polynomials. Graphical plots of the strain energy release rate, crack energy, and crack-opening displacement for the problem in different particular cases are presented.