The Elastic Boundary Value Problem Of Second-Order Homogeneous Linear Ordinary Differential Equation And Its Applied Research

As the relative derivative of the function,elasticity reflects the relative rate of change of the dependent variable to the independent variable.According to its definition,the boundary value problem of second-order linear homogeneous ordinary differential equation under elastic outer boundary condition is established.Based on the related theories of the third kind of boundary value problems of second-order linear homogeneous ordinary differential equation,a similar construction method for solving second-order linear homogeneous ordinary differential equation of elastic boundary value problem is refine,and its concrete steps are summarized.Then it is applied to several typical special equations of elastic boundary value problem,such as: Bessel equation,Modified Bessel equation,Extended Modified Bessel equation,Thomson equation and S-L equation.The similar structures of solutions of the boundary value problems of these special equations are obtained.The influence of the elastic coefficient and the parameters in the fixed solution equation on the solution are analyzed.It lays a solid theoretical foundation for the application of various special equations in engineering and scientific problems.As the practical application of elastic boundary value problem in engineering,the well test analysis of petroleum development engineering is emphatically discussed.According to the elasticity of the pressure difference to the boundary,the elastic outer boundary condition is introduced.The former idealized outer boundary conditions of constant pressure,closed and infinite are regarded as its special cases.Then,under the elastic outer boundary condition,a homogeneous reservoir seepage model considering wellbore storage and skin effect,and a homogeneous reservoir seepage model considering wellbore storage and introducing effective well diameter are established.By means of dimensionless variables and Laplace transformation,the models are transformed into elastic boundary value problems of secondorder linear homogeneous ordinary differential equations in Laplace space.The solutions of the two models in Laplace space are obtained by using the similar construction method.Finally,Laplace numerical inversion technique is used to draw characteristic curves and analyze the influence of elastic coefficients and physical parameters on dimensionless bottomhole pressure.The application of elastic outer boundary condition in petroleum engineering is strongly illustrated,which enlarges the interpretation range of data analysis and improves the accuracy of analysis,and has profound application value.