Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 13

As a simple example of an ODE, we mention the population dynamics model

(1.1.1)

If , then the equation is called homogeneous; otherwise, it is called inhomogeneous. For , the homogeneous equation has an exponentially growing analytic solution given by , where is the initial population. On the contrary, yields a population that vanishes (dies out) with time.

 The order of a differential equation is the order of the highest derivative of the function that appears in the equation.

 If the function depends on more than one variable, and the differential equation possesses derivatives with respect to at least two variables, then the differential equation is called a partial differential equation (PDE), e.g.is a homogeneous PDE of the second order, whereas for , the equationsandare nonhomogeneous PDEs of the second order.

 A solution to a differential equation is a function; (e.g. , , or above), which satisfies the corresponding differential equation.

 In general, the solution of a differential equation cannot be expressed in terms of elementary functions, and numerical methods are the only way to solve the differential equations through constructing approximate solutions. Then, the main questions areTo what extent does the approximate solution preserve the physical properties of the exact solution, or satisfies a corresponding, discrete, version of the differential equation (consistency)?How sensitive is the solution to the change of the data (stability)?How close is the approximate solution to the exact solution (convergence)?Which are the adequate environments to measure this closeness?

 These are some of the questions that we want to deal within this text when approximating with the FEMs.

 A linear ODE of order has the general form:where denotes the derivative, with respect to , and , with (the ‐th order derivative). The corresponding linear differential operator is denoted by