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 "Boundary Value Solution" Fourth-Order Linear Differential Equation
 X Y''
 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.013877 0.049863 0.102318 0.165629 0.23421 0.302527 0.365102 0.416539 0.451532 0.464886

 (Suggested 10 - 100)

IMPLEMENTATION
Finite Difference Method (High-Order Equation)

We have shown the finite difference method to second-order differential equations. In fact, the finite difference method is a very general numerical method for solving various types of differential equations. It can easily be applied to higher-order differential equations.

Consider the general fourth-order linear equation:

 y(4) = P(x)y + Q(x)

Two boundary conditions are specified at each end of the solution domain [a, b].

 Type I: y(a) = ya, y'(a) = va or Type II: y'(a) = va, y'''(a) = ga

Many sets of boundary conditions are possible. However, one must take care when specifying boundary conditions, because some combinations of boundary conditions may not work for the fourth-order finite difference method.

For example, the combination of y(a) and y'''a or y'(a) and y''a will not work.

It is possible to solve the fourth-order linear differential equation using the method we implemented under the second-order differential equation shown before. See "Second-Order Finite Differential Equations".

Consider the following equation

 y(4) = y sin x + x, y(0) = y'(0) = 0, y'(1) = 0, y'''(1) = -5

so,

 P(x) = sin x, Q(x) = x

Running this example produces the results shown above. The accuracy is improved by increasing the number of mesh points. As before, we feel the user has better participation by running real-time instead of showing samples of its implementation.The reader can try variations above.

Testing the FiniteDifferenceLinear4 Method

In order to test the FiniteDifferenceLinear4 method as defined above, a new TestFiniteDifferenceLinear4() static method has been added and executed. All supporting code and methods are not shown.

static void TestFiniteDifferenceLinear4();
{
BoundaryValue bv = new BoundaryValue();
string strScore;
int intScore;
strScore = txtInput.Text;
intScore = Int32.Parse(strScore);
bv.xa = 0.0;
bv.xb = 1.0;
bv.ya = 0.0;
bv.va = 0.0;
bv.vb = 0.0;
bv.gb = -5;
bv.n = intScore;
ListBox1.Items.Clear();
ListBox2.Items.Clear();
double[] x;
VectorR y = bv.FiniteDifferenceLinear4(f5, out x);
for (int i = 0; i < x.Length; i++)
{
double t2 = x[i];
t1 = double.Parse(t1.ToString("#0.######"));
double t2 = y[i];
t2 = double.Parse(t2.ToString("#0.######"));
}
}

static VectorR f5(double x);
{
VectorR result = new VectorR(2);
result[0] = Math.Sin(x);
result[1] = x;
return result;
}

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