An Introduction to the Finite Element Method for Differential Equations. Mohammad Asadzadeh

An Introduction to the Finite Element Method for Differential Equations

rel="nofollow" href="#fb3_img_img_1a2a4e30-523e-5098-bf3e-bbc0e2b474d0.png" alt="images"/> are any constants, then relation (1.4.2) is fulfilled. This is an immediate consequence of the fact that (1.4.1) and (1.4.2) are valid for images

replaced with the derivative of any admissible order. A linear differential equation defines a linear differential operator: the equation can be expressed as images

, where

is a linear operator and images

is a given function. The differential equation of the form images

is called a homogeneous equation. For example, define the operator images

. Then

is a homogeneous equation, while the equation

is an example of an inhomogeneous equation. In a similar way, we may define another type of constraint for the PDEs that appears in many applications: the boundary conditions. In this regard, the linear boundary conditions are defined as operators images satisfying

(1.4.4) equation

at the boundary of a given domain images .

Note that Laplace, heat, and wave equations are linear. Likewise, all the important boundary conditions (Dirichlet, Neumann, Robin) are linear.

The Superposition Principle. An important property of the linear operators is that if the functions images , satisfy the linear differential equations images and the linear boundary conditions images for images , then any linear combination images , satisfies the equation images as well as the boundary condition images . In particular, if each of the functions images , satisfies the homogeneous equation images and the homogeneous boundary condition images , then every linear combination of them satisfies that equation and boundary condition too. This property is called the superposition principle. It allows to construct complex solutions through combining simple solutions: suppose we want to determine all solutions of a differential equation associated with a boundary condition, viz.

(1.4.5) equation

We consider the corresponding, simpler homogeneous problem:

(1.4.6) equation

Now, it suffices to find just one solution, say images of the original problem (1.4.5). Then, for any solution images of (1.4.5), images satisfies (1.4.6), since images and images . Hence, we obtain a general solution of (1.4.5) by adding the general (homogeneous) solution images of (1.4.6) to any particular solution of (1.4.5).

Following the same idea, one may apply superposition to split a problem involving several inhomogeneous terms into simpler ones each with a single inhomogeneous term. For instance, we may split (1.4.5) as

and then take images .

The most important application of the superposition principle is in the homogeneous case: linear homogeneous differential equations satisfying homogeneous boundary conditions (which we repeat from above).

The Superposition principle for the homogeneous case. If the functions images , satisfy (1.4.6): the linear differential equation images and the boundary conditions (linear) images for images , then any linear combination images , satisfies the same equation and boundary condition: (1.4.6).

1.4.1 Exercises

1 Problem 1.1 Consider the problemClearly, the function is a solution. Is this solution unique? Does the answer depend on ?

2 Problem 1.2 Consider the problemIs the solution unique? ( is a given function).Under what condition on a solution exists?

3 Problem 1.3 Suppose are solutions of the linear differential equation , where . Under what condition on the constant coefficients is the linear combination also a solution of this equation?

4 Problem 1.4 Consider the nonlinear ODE .Show that and both are solutions, but is not a solution.For which value of is a solution? What about ?

5 Problem 1.5 Show that each of the following equations has a solution of the form for a proper choice of constants . Find the constants for each example.

6 Problem

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