Mixed Boundary Conditions In Solving Differential Equations

Akash Srivastava

RC1802A04

Reg No. 10802311

Abstract:

A general system of the time-dependent partial differential equations containing several arbitrary initial and boundary conditions is considered. A hybrid method based on artificial neural networks, minimization techniques and collocation methods is proposed to determine a related approximate solution in a closed analytical form. The optimal values for the corresponding adjustable parameters are calculated. An accurate approximate solution is obtained, that works well for interior and exterior points of the original domain. Numerical efficiency and accuracy of the hybrid method are investigated by two-test problems including an

initial value and a boundary value problem for the two-dimensional biharmonic equation.

Introduction:

In mathematics, a mixed boundary condition for a partial differential equation indicates that different boundary conditions are used on different parts of the boundary of the domain of the equation.

For example, if u is a solution to a partial differential equation on a set Ω with piecewise-smooth boundary ∂Ω, and ∂Ω is divided

into two parts, Γ₁ and Γ₂, one can use a Dirichlet boundary condition

on Γ₁ and a Neumann boundary

condition on Γ₂:

where u₀ and g are given functions defined on those portions of the boundary.

Robin boundary condition is another type of hybrid boundary condition; it is a linear combination of Dirichlet and Neumann boundary conditions.

Differential Equations

A differential equation is an equation which contains a derivative (such as dy/dx).

Example

To verify that something solves an equation, you need to substitute it into the equation and show that you get zero. So, here we need to work out dy/dx and show that this

is equal to the right hand side when we substitute the x-3 into it.

dy/dx = -3x-4

and check that when you substitute x-3 into the right hand side you also get -3x-4.

Solving

When given a differential equation, you will often be asked to "solve" the differential equation or find the "general solution". This basically means find an expression which does not contain any derivatives. To do this you will need to integrate.

Example

Boundary value problem:

Shows a region where a differential equation is valid and the associated boundary values

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.

To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.

Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.

Initial or Boundary Conditions:

You may be given information which allows you to work out the constant of integration.

In the above example, for example, you may have been told that y(0) = 0 [which means y = 0 when x = 0].

We can substitute these values into our answer to determine c:

0 = 0 + 0 + c, so c = 0

The answer in this case would therefore be y = 5x2/2 + 3x .

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.

To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.

Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.

If the problem is dependent on both space and time, then instead of specifying the value of the problem at a given point for all time the data could be given at a given time for all space. For example, the temperature of an iron bar with one end kept at absolute zero and the other end at the freezing point of water would be a boundary value problem.

Concretely, an example of a boundary value (in one spatial dimension) is the problem

to be solved for the unknown function y(x) with the boundary conditions

Without the boundary conditions, the general solution to this equation is

From the boundary condition y(0) = 0 one obtains

which implies that B = 0. From the boundary condition y(π / 2) = 2 one finds

and so A = 2. One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is

Types of boundary value problems:

The boundary value problem for an idealised 2D rod

If the boundary gives a value to the normal derivative of the problem then it is a Neumann boundary condition. For example, if there is a heater at one end of an iron rod, then energy would be

added at a constant rate but the actual temperature would not be known.

If the boundary gives a value to the problem then it is a Dirichlet boundary condition. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.

If the boundary has the form of a curve or surface that gives a value to the normal derivative and the problem itself then it is a Cauchy boundary condition.

Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an elliptic operator, one discusses elliptic boundary value problems. For an hyperbolic operator, one discusses hyperbolic boundary value problems. These categories are further subdivided into linear and various nonlinear types.

REFERENCE:

1)http://en.wikipedia.org/

wiki/Differential_equation

2)http://www.sosmath.com/

diffeq/diffeq.html

3)http://en.wikipedia.org/

wiki/Boundary_value_problem

4)http://en.wikipedia.org/

wiki/Mixed_boundary_condition

5)http://en.wikipedia.org/

wiki/Boundary _value_problem