Abstract:
: In this work, the effect of radiation on heat transfer of an electrically conducting fluid flow over a linearly shrinking surface subject to heat sink and magnetic field applied normal to the plane of the flow is investigated analytically. The governing boundary layer equations for fluid flow and energy are reduced into ordinary differential equations by means of a similarity transformations. Closed form exact solutions of the reduced energy equation have been obtained for both prescribed power-law surface temperature (PST)and power-law wall heat flux (PHF) boundary conditions and these solutions are valid for all M > 1, where M is the magnetic interaction parameter. It is found that the temperature within the fluid is reduced significantly with the increasing values of radiation parameter, Prandtl number, heat sink and magnetic field parameters for both PST and PHF cases. Some solutions involving negative temperature values are also noticed. In some cases, temperature overshoot near the wall is also observed.

Abstract:
The boundary layer flow and heat transfer over a permeable sheet axisymmetrically shrinking with velocity inversely proportional to the radial distance, is investigated subject to suction at the surface. The suction at the sheet is assumed to be inversely proportional to the radial distance. The governing partial differential equations for boundary layer flow and heat transfer are reduced into ordinary differential equations (ODEs) by a similarity transformation. The reduced ODEs are then solved numerically by finite element method for power-law temperature boundary conditions. It is found that radial velocity is decreased with the increase in suction at the surface. It is also observed that the thermal boundary layer thickness decreases with the increase in suction parameter and Prandtl number. For some values of power law index, temperature overshoot is observed. Some solutions involving negative non-dimensional temperature values are also noticed.

Abstract:
In this paper, an axisymmetric flow and heat transfer of electrically conducting fluid over a nonlinearly shrinking surface in the presence of magnetic field and suction at the surface is investigated numerically. The shrinking velocity as well as suction at the sheet are assumed to follow the power law of radial distance. A magnetic field is applied normal to the sheet. The governing boundary layer PDEs in cylindrical polar coordinates are reduced into highly nonlinear ordinary differential equations by a similarity transformation. The reduced ODEs are then solved numerically by finite element method for power-law temperature boundary conditions. It is found that radial velocity is decreased with the increase in magnetic field strength and suction at the surface. It is also observed that the thermal boundary layer thickness decreases with the increase in magnetic field strength, suction parameter, power-law index for shrinking velocity and Prandtl number. On the other hand, the thermal boundary layer thickness is increased for increasing values of power-law index for temperature.

Abstract:
Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: $ e^{-\alpha p} x e^{\alpha p} = x+ i \alpha $, to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.

Abstract:
The modified factorization technique of a quantum system characterized by position-dependent mass Hamiltonian is presented. It has been shown that the singular superpotential defined in terms of a mass function and a excited state wave function of a given position-dependent mass Hamiltonian can be used to construct non-singular isospectral Hamiltonians. The method has been illustrated with the help of a few examples.

Abstract:
We use supersymmetry transformations to design transparent and one-way reflectionless (thus unidirectionally invisible) complex crystals with balanced gain and loss profiles. The scattering coefficients are investigated using the transfer matrix approach. It is shown that the amount of reflection from the left can be made arbitrarily close to zero whereas the reflection from the right is enhanced arbitrarily (or vice versa).

Abstract:
We address the existence and stability of spatial localized modes supported by a parity-time-symmetric complex potential in the presence of power-law nonlinearity. The analytical expressions of the localized modes, which are Gaussian in nature, are obtained in both (1+1) and (2+1) dimensions. A linear stability analysis corroborated by the direct numerical simulations reveals that these analytical localized modes can propagate stably for a wide range of the potential parameters and for various order nonlinearities. Some dynamical characteristics of these solutions, such as the power and the transverse power-flow density, are also examined.

Abstract:
In this work, exact mathematical functions have been formulated for three important theoretical Shruti (micro tonal interval) distributions, i.e. for Western Compilation, Deval, and Nagoji Row in Hindustani music. A generalized mathematical function for Shrutis has also been formulated. This generalized function shows a very high order of conformity with the experimentally derived Shruti distribution, than those of the theoretical Shruti distributions.