Urysohn integral equation is one of the most applicable topics in both pure and applied mathematics. The main objective of this paper is to solve the Urysohn type Fredholm integral equation. To do this, we approximate the solution of the problem by substituting a suitable truncated series of the well known Legendre polynomials instead of the known function. After discretization of the problem o...

Abstract: The non-zero unique integer solutions to the quadratic Diophantine equation with three unknowns 2 x  14 xy y  z are examined. We derive integral in four different patterns. A few intriguing relationships between answers and a polygonal integers shown

In this paper, quadratic and sextic B-splines are used to construct an approximating function based on the integral values instead of the function values at the knots. This process due to the type of used B-splines (fourth order or sixth order), called integro quadratic or sextic spline interpolation. After introducing the integro quartic and sextic B-spline interpolation, their convergence is ...

in this paper, we will compare a homotopy perturbation algorithm and taylor series expansin method for a system of second kind fredholm integral equations. an application of he’s homotopy perturbation method is applied to solve the system of fredholm integral equations. taylor series expansin method reduce the system of integral equations to a linear system of ordinary differential equation.

In this study we construct new integral representations of Jost-type solutions the quadratic pencil Sturm-Liouville equation with piece-wise constant coefficient on entire real line. Our aim is to express special in form some operators which kernels related potential function equation. This problem technically diffucult due discontinous causes kernel also have a jump discontinuity.

in this paper, we solve the quadratic $alpha$-functional equations $2f(x) + 2f(y) = f(x + y) + alpha^{-2}f(alpha(x-y)); (0.1)$ where $alpha$ is a fixed non-archimedean number with $alpha^{-2}neq 3$. using the fixed point method and the direct method, we prove the hyers-ulam stability of the quadratic $alpha$-functional equation (0.1) in non-archimedean banach spaces.

aSchool of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China; bCollege of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang 050061, People’s Republic of China; cKey Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, People’s Republic...

In the present paper we outline the stochastic limit approach to superfluidity. The Hamiltonian describing the interaction between the Bose condensate and the normal phase is introduced. Sufficient in the stochastic limit condition of superfluidity is proposed. Existence of superfluidity in the stochastic limit of this system is proved and the non– linear (quadratic) equation of motion describi...