// Expand N(x)/D(x) using partial fractions.
private static Expression ExpandPartialFractions(Expression N, Expression D, Expression x)
{
List <Expression> terms = new List <Expression>();
List <Variable> unknowns = new List <Variable>();
List <Expression> basis = new List <Expression>();
foreach (Expression i in Product.TermsOf(D))
{
// Get the multiplicity of this basis term.
Expression e = i;
int n = Power.IntegralExponentOf(e);
if (n != 1)
{
e = ((Power)i).Left;
}
// Convert to a polynomial.
Polynomial Pi = Polynomial.New(e, x);
// Add new terms for each multiplicity n.
for (int j = 1; j <= n; ++j)
{
// Expression for the unknown numerator of this term.
Expression unknown = 0;
for (int k = 0; k < Pi.Degree; ++k)
{
Variable Ai = Variable.New("_A" + unknowns.Count.ToString());
unknown += Ai * (x ^ k);
unknowns.Add(Ai);
}
terms.Add(Product.New(unknown, Power.New(e, -j)));
}
basis.Add(i);
}
// Equate the original expression with the decomposed expressions.
D = Sum.New(terms.Select(j => (Expression)(D * j))).Expand();
Polynomial l = Polynomial.New(N, x);
Polynomial r = Polynomial.New(D, x);
// Equate terms of equal degree and solve for the unknowns.
int degree = Math.Max(l.Degree, r.Degree);
List <Equal> eqs = new List <Equal>(degree + 1);
for (int i = 0; i <= degree; ++i)
{
eqs.Add(Equal.New(l[i], r[i]));
}
List <Arrow> A = eqs.Solve(unknowns);
// Substitute the now knowns.
return(Sum.New(terms.Select(i => i.Evaluate(A))));
}
/// <summary>
/// Partially solve a linear system of differential equations for y[t] in terms of y[t - h]. See SolveExtensions.PartialSolve for more information.
/// </summary>
/// <param name="f">Equations to solve.</param>
/// <param name="y">Functions to solve for.</param>
/// <param name="t">Independent variable.</param>
/// <param name="h">Step size.</param>
/// <param name="method">Integration method to use for differential equations.</param>
/// <returns>Expressions for y[t].</returns>
public static List <Arrow> NDPartialSolve(this IEnumerable <Equal> f, IEnumerable <Expression> y, Expression t, Expression h, IntegrationMethod method)
{
// Find y' in terms of y.
List <Arrow> dy_dt = f.Solve(y.Select(i => D(i, t)));
// If dy/dt appears on the right side of the system, the differential equation is not linear. Can't handle these.
if (dy_dt.Any(i => i.Right.DependsOn(dy_dt.Select(j => j.Left))))
{
throw new ArgumentException("Differential equation is singular or not linear.");
}
return(NDIntegrate(dy_dt, t, h, method)
.Select(i => Equal.New(i.Left, i.Right))
.PartialSolve(y));
}
/// <summary>
/// Solve a linear system of differential equations with initial conditions using the laplace transform.
/// </summary>
/// <param name="f"></param>
/// <param name="y"></param>
/// <param name="y0"></param>
/// <param name="t"></param>
/// <returns></returns>
public static List <Arrow> DSolve(this IEnumerable <Equal> f, IEnumerable <Expression> y, IEnumerable <Arrow> y0, Expression t)
{
// Find F(s) = L[f(t)] and substitute the initial conditions.
List <Equal> F = f.Select(i => Equal.New(
L(i.Left, t).Evaluate(y0),
L(i.Right, t).Evaluate(y0))).ToList();
// Solve F for Y(s) = L[y(t)].
List <Arrow> Y = F.Solve(y.Select(i => L(i, t)));
// Take L^-1[Y].
Y = Y.Select(i => Arrow.New(IL(i.Left, t), IL(i.Right, t))).ToList();
if (Y.DependsOn(s))
{
throw new Exception("Could not find L^-1[Y(s)].");
}
return(Y);
}