The main purpose of this paper is to consider an explicit form of the Padé-type operators. To do so, we consider the representation of Padé-type approximants to the Fourier series of the harmonic functions in the open disk and of the $L^p$-functions on the circle by means of integral formulas, and, then we define the corresponding Padé-type operators. We are also oncerned with the properties of these integral operators and, in this connection, we prove some convergence results.

We consider series solutions for the location of the optimal exercise boundary of an American option close to expiry. By using Monte Carlo methods, we compute the expected value of an option if the holder uses the approximate location given by such a series as his exercise strategy, and compare this value to the actual value of the option. This gives an alternative method to evaluate approximations. We find the series solution for the call performs excellently under this criterion, even for large times, while the asymptotic approximation for the put is very good near to expiry but not so good further from expiry.

We investigate the problem of free convection heat transfer near an isothermal stretching sheet. This has been done under the simultaneous action of buoyancy, radiation, and transverse magnetic field. The governing equations are solved by the shooting method. The velocity and temperature functions are represented graphically for various values of the flow parameters: radiation parameter $F$, free convection parameter $\text{Gr}$, magnetic parameter $M$, Prandtl number $\text{Pr}$, and the parameter of relative difference between the temperature of the sheet, and the temperature far away from the sheet $r$. The effects of the radiation and magnetic field parameters on the shear stress and heat flux are discussed.

We establish a blowup result to an initial boundary value problem for the nonlinear wave equation $u_{tt}-M\left(\Vert B^{1/2}u\Vert {}^2\right)Bu+k{u}_t=\left|u\right|{}^{p-2}$, $x\in \Omega$, $t>0$.