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Short Assignment Requirements

I need the analytical and numerical solution with the programming code

Assignment Description

Trabajo computacional 1: Entregar 4 de junio

Parte teo´rica:

1.    (Parte A: Para aquellos que nunca han llevado un curso de parciales.) Determine la soluci´on de los siguientes problemas usando separaci´on de variables y por transformada de Fourier, segu´n convenga. (a)










2.    Parte B: para los alumnos inscritos en el curso. Problemas 2.3, 2.5 y2.7 del Morton y Mayers. (30 puntos)

3.    Parte C: hacer un programa que resuelva num´ericamente con el m´etodoθ para θ = 0,1/2,1, la ecuacio´n del calor para distintas condiciones de frontera. El sistema de ecuaciones debe resolverse por Thomas. Obtenga resultados con una precisio´n de dos cifras decimales. Justifique su seleccio´n de h y ∆t. Si el problema admite soluci´on estacionaria, termine las iteraciones en el tiempo cuando alcance la soluci´on estacionaria.

(a)     Calibre su programa resolviendo num´ericamente los ejercicios 2, 3 y 5. Contraste su solucio´n num´erica con la exacta.

(b)    Seleccione dos de los siguientes problemas:

i. (obligatorio para Fernando) El valor de una opci´on put europea con vencimiento al tiempo T y con precio de ejercicio K satisface el siguiente problema en EDP:


donde S(t) es precio del subyacente con volatilidad σ > 0 y con tasa de inter´es r > 0. Las condiciones iniciales y de frontera son las siguientes:

                                              u(S,T) = Max{K S,0}                     U(0,t) = Ker(tt),

lim u(S,t)       =          0. S→∞

A.      Determinar la solucio´n anal´ıtica de este problema pormedio de la Transformada de Fourier. Seguir los pasos indicados en P. Wilmott, S. Howinson y J. Dewynne. The Mathematics of financial derivatives. Cambridge University Press. 1995

B.      Determinar la soluci´on num´erica transformando el prob-lema a la solucio´n de la ecuaci´on del calor. Usar el m´etodo de Crank-Nicholson con el algoritmo de Thomas con α =


C.      Determinar la soluci´on para los siguientes datos: T =

6,12 meses, σ = 0.1,0.2, r = 0.0435 y K = 10 con S0 =


ii.       Determinar y graficar la soluci´on anal´ıtica del siguiente prob-lema de advecci´on-difusi´on para los siguientes valores de α = 1,0.1,0.01 y ν = 1.


con condiciones de frontera e iniciales:

                                                                u(0,t) = u(1,t) = 0       u(x,0) = ex/2.

Encontrar la solucio´n anal´ıtica usando el Hint. Resolver el problema anterior por medio de Euler expl´ıcito, Crank-Nicholson y Euler impl´ıcito transformando el problema a la ecuacio´n del calor. Graficar la solucio´n exacta con la num´erica para T = 1.

Hint: La soluci´on) con w solucio´n de la ecuaci´on del calor wt = α2wxx.

iii.    Considere la siguiente ecuacio´n de reaccio´n difusio´n que apareceen la modelacio´n de problemas de cin´etica qu´ımica y en biomatema´ticas. La EDP es de la forma


con condiciones de frontera e iniciales:

                                ;      .

A.      Suponga que f(u) = 6u(1 − u). Admite solucio´n estacionaria?

B.      Aplicar el m´etodo de Euler expl´ıcito con la restriccio´n deestabilidad y al evaluar el t´ermino no-lineal en el tiempo n se obtiene el siguiente sistema no lineal a resolver

F(U~ n+1) = Un+1 − ∆tf(U~ n) − (1 − αA)U~ n = 0.

C.      Evaluar la funci’on no lineal evaluando en el tiempo n+1 para dar lugar al problema no lineal:

F(U~ n+1) = Un+1 − ∆tf(U~ n+1) − (1 − αA)U~ n = 0.

Aplicar el m´etodo de Newton conpara el problema anterior para aproximar la soluci´on hasta alcanzar la soluci´on estacionaria, en caso de que exista

(70 puntos)

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