Petroleum Reservoir Engineering by Non-linear Singular Integral Equations

Multiphase Flows in Oil Reservoir Engineering by Non-linear Singular Integral Equations

Abstract An innovative and groundbreaking mathematical approach is proposed for the determination of the properties of reservoir materials, when oil reserves together with water are moving through porous media. Such problem seems to be very important for oil reservoir engineering. So, the above problem is reduced to the solution of a non-linear singular integral equation, which is numerically e...

Solving singular integral equations by using orthogonal polynomials

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...

Modified homotopy method to solve non-linear integral equations

In this article we decide to define a modified homotopy perturbation for solving non-linear integral equations. Almost, all of the papers that was presented to solve non-linear problems by the homotopy method, they used from two non-linear and linear operators. But we convert a non-linear problem to two suitable non-linear operators also we use from appropriate bases functions such as Legendre ...

Fully non linear equations with integral diffusions

The authors establish weighted decay rates for the energy associated with the Cauchy problem { utt − div(b(x)∇u) + a(x)ut = 0, x ∈ R, t > 0 u(0, x) = u0(x), ut(0, x) = u1(x), x ∈ R. Such a system appears in models for traveling waves in a non-homogeneous gas with damping that changes with the position. This problem has been studied intensively for the homogeneous medium (when b is constant), bu...

A Method to Estimate the Solution of a Weakly Singular Non-linear Integro-differential Equations by Applying the Homotopy Methods