Analysis has been carried out on velocity configuration for Magnetohydrodynamic (MHD) flow in a straight horizontal pipe of elliptical cross section when Reynolds number and half-length of major axis are adjusted. An electrically conducting, viscous and incompressible fluid flow in the z-direction of a long pipe with elliptical cross-section in x-y plane has been scrutinized. An applied magnetic field is conveyed to the pipe in the y-direction. The fluid is driven by Lorentz force and gravitational force. The governing equations are partial differential equations (pdes) formulated in terms of cylindrical coordinates (r, , z) by amalgamating r-component of Navier-Stokes equation, Ohm’s law, equation of continuity and cross-section of pipe. Navier-Stokes equation is non-dimensionalised, composed in terms of stream function, , converted to an ordinary differential equation (ode) using similarity transformation and evaluated numerically using Finite Element Method (FEM). Results are obtained for velocities for values of Reynolds number, Re? 10.0 and half major axis distance, a 0.0034. Tables and graphs are created and sketched respectively using these findings. The eventualities confess that: When Reynolds number and a half distance of major axis are escalated, velocity of the fluid rises at the centre of the pipe but dwindle to zero towards the periphery of the pipe in both cases. Subsequently, it is anticipated that a pipe of elliptical cross-section would be more beneficial in research on natural phenomena and technological applications than a circular one with the same depth. Fluid velocity and capacity will be higher in the elliptical pipe than in circular pipe resulting in higher MHD effects.

We consider the singular Cauchy problem of Euler-Poisson-Darboux equation(EPD) of the form u tt + k t u t = ∇ 2 u u(x, 0) = f (x), u t (x, 0) = 0

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A general exact solution to the n-dimensional regular Cauchy problem of Euler-Poisson-Darboux (EPD) equation has been studied. Firstly, the general exact solution for the one dimensional regular Cauchy problem of EPD has been worked out. The EPD which is a second order Partial Differential Equation (PDE) is converted into an Ordinary Differential Equation (ODE) by method of separation of variables. On solving the ODE, the first complementary function (cf) is obtained directly. The second cf is obtained when the first derivative is eliminated from the ODE and then the ODE solved. When the expression for eliminating the first derivative is solved, a third term is obtained. The general solution for the one dimensional regular Cauchy EPD is therefore the product of the three terms. The procedure has been repeated for the two dimensional and n-dimensional cases. The general solutions for these cases are products of four terms and n+2 terms respectively. Finally, the general exact solution for n-dimensional regular Cauchy wave equation when k = 0, has also been obtained.

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