A New Method and Applications of the Boundary Value Problem of Differential Equation

2014 ◽  
Vol 937 ◽  
pp. 695-699
Author(s):  
Hong E Li ◽  
Xiao Xu Dong ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Cong Yin Fan
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Kernel Function ◽  
Boundary Value ◽  
Outer Boundary ◽  
Seepage Flow ◽  
Petroleum Engineering ◽  
Constructive Method ◽  
Linear Homogeneous Differential Equation ◽  
The Mathematical Model

The similar structure of solution for the boundary value problem of second order linear homogeneous differential equation has been studied. Based on the analysis of the relationship between similar structure of solution, its kernel function, the equation and boundary conditions, similar constructive method (shortened as SCM) of solution is obtained. According to the SCM, the similar structure of solution and its kernel function are constructed for the mathematical model of homogeneous reservoir which considers the influence of bottom-hole storage and skin effect under the infinite outer boundary condition. The SCM is a new and innovative way to solve boundary value problem of differential equation and seepage flow theory, which is especially used in Petroleum Engineering.

Similar Constructing Method for Solving the Model of the Plane Radial Flow of Dual Permeability Reservoir

2013 ◽  
Vol 419 ◽  
pp. 43-50 ◽  
Author(s):  
Cong Yin Fan ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Ming Hu ◽  
Hui Chun Li
Keyword(s):  
Boundary Value Problem ◽  
Radial Flow ◽  
Boundary Value ◽  
Outer Boundary ◽  
Structural Formula ◽  
Constructive Method ◽  
Well Bore ◽  
Bottom Hole ◽  
Constant Rate ◽  
Seepage Model

A seepage model of producing at a constant rate, based on the three outer boundary (infinite, closed, constant value) conditions and regardless of the well-bore storage and skin effects, is established for the problem of the plane radial flow of dual permeability reservoirs. Firstly, we get dimensionless model by introducing the dimensionless variables. Second, we obtain a boundary value problem of ordinary differential equation in the Laplace space by using the Laplace transformation. Finally, we prove that the solution to the boundary value problem has similar structural formula. Therefore a new method for solving such seepage model is obtainedSimilar Constructive Method (shortened as SCM). And then, according to the modified numerical inversion formula of Stehfest, we draw the curves of bottom-hole under the three kinds of outer boundary conditions by using MATLAB.

scholarly journals Similar Construction Method of Solution for Solving the Mathematical Model of Fractal Reservoir with Spherical Flow

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Cui-Cui Sheng ◽  
Jin-Zhou Zhao ◽  
Yong-Ming Li ◽  
Shun-Chu Li ◽  
Hu Jia
Keyword(s):  
Boundary Conditions ◽  
Boundary Value ◽  
Outer Boundary ◽  
Petroleum Engineering ◽  
Constructive Method ◽  
Porous Media Model ◽  
Method Of Solution ◽  
Complex Boundary ◽  
Linear Differential ◽  
Set Up

On the basis of similar structure of solution for a second-order linear differential equation's boundary value problem, and our analysis of the relationship between this similar structure and its kernel function, the differential equation, and the boundary conditions, we propose a new simple solution—similar constructive method of solution (SCMS)—and sum up its detailed steps. We set up a porous media model under three kinds of outer boundary conditions (infinite, constant pressure, and closed), in which the influences of fractal dimension, spherical flow, skin effect, and storage are taken into consideration. And then we use SCMS to solve it. The SCMS is a straightforward method for differential equation's boundary value problems with complex boundary conditions, especially for solving the reservoir models in petroleum engineering.

Similar Constructive Method of Solution for the Model of Nonlinear Spherical Seepage in Composite Reservoir

2014 ◽  
Vol 700 ◽  
pp. 597-601
Author(s):  
Feng Jiu Zhang ◽  
Xu Xia Xiao ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Qiang Wang
Keyword(s):  
Flow Model ◽  
Boundary Value ◽  
Outer Boundary ◽  
Seepage Flow ◽  
Analytic Solutions ◽  
Flow Equation ◽  
Constructive Method ◽  
Gradient Term ◽  
Transform Method ◽  
Method Of Solution

The nonlinear spherical seepage flow model has been established for the composite reservoir model. The nonlinear spherical seepage flow model considers the well produce at a constant rate, and the quadratic gradient term under three outer boundary conditions (closed, constant pressure and infinite). Firstly, through variable substitutions, the seepage flow equation is linearized; then the model is transformed into the boundary value problem of an ordinary differential equation by employing the Laplace transform method. It has been confirmed that the Laplace space analytic solutions of such boundary value problems has a formula under different external boundaries, using the Similar Constructive Method(it is a simple and effective new idea for solving this class seepage flow model, complicated calculus calculation is avoided). The prospect of this new method is promising for understanding and studying the inherent laws of fluids flow.

scholarly journals A Study on Solving the Seepage Flow Model of Three-Area Composite Reservoir

2014 ◽  
Vol 670-671 ◽  
pp. 599-603 ◽  
Author(s):  
Xiao Xu Dong ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Feng Jiu Zhang
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Flow Model ◽  
Constant Pressure ◽  
Outer Boundary ◽  
Seepage Flow ◽  
Well Bore ◽  
Analysis Software

This paper studies the seepage flow mathematical model of three-area composite reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary), in which influences of well-bore storage and skin factor are not taken into consideration. On the basic of theory of similar structure of solution of boundary value problem of differential equation, this paper obtain the solution of the seepage flow model of three-area composite reservoir. The study is not only conducive to further analyze the inherent law of the solution and solve corresponding application problems, but also easy to compile corresponding analysis software.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

scholarly journals Second-order neutrosophic boundary-value problem

2021 ◽  
Author(s):  
Sandip Moi ◽  
Suvankar Biswas ◽  
Smita Pal(Sarkar)
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Differential Calculus ◽  
Boundary Value ◽  
Second Order ◽  
Numerical Examples ◽  
Different Types ◽  
First Time

AbstractIn this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation.

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

1997 ◽  
Vol 4 (6) ◽  
pp. 557-566
Author(s):  
B. Půža
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Vector Function ◽  
Unique Solvability ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Deviating Arguments ◽  
Measurable Functions

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

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