As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ord...As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point <em>t</em> = 0 and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.展开更多
A new approach is given to analyse the regularity of solutions near singular points for the interface problems of second order elliptic partial differential equations. For general equations with nonsymmetric dominant ...A new approach is given to analyse the regularity of solutions near singular points for the interface problems of second order elliptic partial differential equations. For general equations with nonsymmetric dominant terms and discontinuous piecewise smooth coefficients, it is proved that solutions in H 1 can be docomposed into two parts, one of which is a finite sum of particular solutions to the corresponding homogeneous equations with piecewise constant coefficients, and the other one of which is the regular part. Moreover a priori estimations are proven.展开更多
文摘As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point <em>t</em> = 0 and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.
文摘A new approach is given to analyse the regularity of solutions near singular points for the interface problems of second order elliptic partial differential equations. For general equations with nonsymmetric dominant terms and discontinuous piecewise smooth coefficients, it is proved that solutions in H 1 can be docomposed into two parts, one of which is a finite sum of particular solutions to the corresponding homogeneous equations with piecewise constant coefficients, and the other one of which is the regular part. Moreover a priori estimations are proven.
