Special Trans Function

Calculates the analytical or numerical solution of Special Trans Function

Special Trans Function (STF) is the exact analytical closed-form solution of transcendental equation:

Psi = U.*exp(-Psi) U(ζ)∈R+.

For the fixed variable x, STF takes the form of analytic expression:

Psi(x,U) = U.*(symsum((U.^n.*(x-n).^n)/factorial(n),n,[0 floor(x)]))./symsum((U.^n.*(x+1-n).^n)/factorial(n),n,[0 floor(x+1)]);

Psi = STF(x,U) calculates the analytical solution of STF by invoking STF1.m (faster than symbolic/limit computation) or STF2.m (fastest).

Psi = STF(U) solves the numerical solution of STF by invoking STF3.

[Psi,xv,xs] = STF(U) invokes STF4.m to solve the numerical solution of STF and the vectorial STF brabch xv around start point xs.

Inputs:

- x: The STF branch, a non-negative scalar value within [0,20]. For double-precision arithmetic, the analytical solution of STF is exact when x∈[0,20] since factorial(n) is exact when n ≤ 21.

- U: A vector of non-negative numbers, U(ζ)∈[0,inf).

- options: defaultopts = struct('Solution','Analytical','Speed','fast'),

struct('Solution','Numerical','Speed','faster'),

Output:

- Psi: The analytical/numerical solution of STF.

- xv: The numerical solution of vectorial STF brabch around x

Example:

x = 2.5;

U = 0:10;

STF(U)

Psi = STF(U)

[Psi,xv] = STF(U)

STF(x, U)

Psi = STF(x, U)

[Psi,xv] = STF(x, U)

opts = struct('Solution','Analytical','Speed','fast');

STF(x, U, opts)

Psi = STF(x, U, opts)

[Psi,xv] = STF(x, U, opts)

opts = struct('Solution','Analytical','Speed','faster');

STF(x, U, opts)

Psi = STF(x, U, opts)

[Psi,xv] = STF(x, U, opts)

opts = struct('Solution','Numerical');

STF(x, U, opts)

Psi = STF(x, U, opts)

[Psi,xv] = STF(x, U, opts)

References:

[1] Perovich SM, Simic SK, Tosic D V, Bauk SI. On the analytical solution of some families of transcendental equations. Appl Math Lett 2007;20:493–8. https://doi.org/10.1016/j.aml.2006.06.008.

Author: Xiankun Gao, [e-mail]