Integral Solutions of the Ternary Cubic Equation 6(x^2+y^2 )-11xy=288z^3

The Ternary cubic Diophantine Equation represented by૟(࢞ ࢟ + ૛ ࢠૡૡ = ૛࢟࢞૚૚ − (૛ ૜ is analyzed for its infinite number of non-zero integral solutions. A few interesting among the solutions are also discussed. Keywords: Diophantine equation, Integral solutions, cubic equation with three unknowns, Ternary equation.

On the Diophantine equations z2 = k(k2 + 3) and z2 = k(k2 + 12)

This paper provides an analytical method of finding all the (positive, integral) solutions of the Diophantine equation z2 = k(k2+3). We also prove analytically that the Diophantine equation z2 = k(k2+12) has no positive, integer solution. J. Bangladesh Acad. Sci. 45(1); 127-129: June 2021

A GENERALISATION OF THE DIOPHANTINE EQUATION x^2+8∙7^b=y^2r

We investigate the integral solutions to the Diophantine equation where . We first generalise the forms of and that satisfy the equation. We then show the general forms of non-negative integral solutions to the equation under several conditions. We also investigate some special cases in which the integral solutions exist.

On pairs of quadratic forms in five variables

In this paper, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given in [H. Iwaniec and R. Munshi, The circle method and pairs of quadratic forms, J. Théor. Nr. Bordx. 22 (2010) 403–419].

POTENTIAL PART OF ELECTRIC FIELD OF ICP COIL

ICP devices are widespread in modern technology, so the capacitive coupling in ICP is well known. It does not make a sufficient impact to the power deposition in plasma, and mostly it assumed to be a harmful effect [1]. Fortunately, making use of Faraday shield allows one to avoid a detail investigation of the capacitive coupling. However, in some cases it would be useful to account both coupling types of the inductor with a plasma [2]. There are many ways for analytical and numerical simulation of the inductors and their coupling with a plasma. In the most simulating tools, the problem comes to FEM PDE model, which includes as minima the body of an inductor wire and the surrounding space. In the paper, we use some other approach, based on integral solutions for electrodynamics potentials. Electric fields of some inductors are studied in the framework of the approach.

Integral solutions of nondense impulsive conformable-fractional differential equations with nonlocal condition

In this paper, a class of nondense impulsive differential equations with nonlocal condition in the frame of the conformable fractional derivative is studied. The abstract results concerning the existence, uniqueness and stability of the integral solution are obtained by using the extrapolation semigroup approach combined with some fixed point theorems.