partial differential equation (PDE) assignment for 40$

MATH 3150 PDE’s for Engineers

Homework 4

1. Let u(x, t) be a solution to the following problem

ut = uxx Q = {(x, t)|0 < x < π, 0 < t < T} , u(x, 0) = sin2 x 0 ≤ x ≤ π, u(0, t) = 0 = u(π, t) 0 ≤ t ≤ T.

(a) Show that 0 < u(x, t) < 1 for (x, t) ∈ Q (b) Show that 0 < u(x, t) < e−t sinx for (x, t) ∈ Q.

Hint: Show that v(x, t) = e−t sinx is a solution to the same equation with the same boundary condition but with different initial condition.

2. Solve heat equation ut = 2uxx for 0 < x < π and t > 0 with following boundary and initial conditions:

u(0, t) = 0 = u(π, t) u(x, 0) = x2 − π2

3. (a) Solve the following problem

ut = auxx − cu, a, c > 0, 0 < x < 1, t > 0 u(x, 0) = f(x), ux(0, t) = 0 = ux(1, t)

(b) Compute the solution for f(x) = cos2 πx

4. Solve the following problem

ut = 25uxx, 0 < x < 2, t > 0 u(x, 0) = x2, 0 < x < 2, ux(0, t) = 0 = u(2, t), t > 0

5. Solve the following problem

ut = 7uxx, −π < x < π, t > 0 u(x, 0) = cosx, −π < x < π,

when it is known that u(x, t) is periodical in x. Note: The fact that u(x, t) is periodical means that u(−π, t) = u(π, t) , and consequently ux(−π, t) = ux(π, t), for t > 0.

6. The following questions will not be graded or checked, that is for a practice only. However, it can be submitted for up to 10% more points in this homework. Partial credits will be given for partial submission. For example if you do the most you will receive all 10%, if you do about a half of it - you will receive extra 5%.

• Section 13.3 questions 1,2,3,5,6. • Section 13.4 1

762 Chapter 13 Method of Separation of Variables

These steps should be understood. riot memorized. It is important to note that

1. The principle of superposition applies to solutions of the PDE (do not add up solutions of various different ordinary differential equations).

2. Do not apply the initial condition u(x, 0) = f(x) until after the principle of superposition.

Problems

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Orrx art 3isin — — sinI. Forthe foJlowing partial differeutial equations,what mdi- (a) rife 0) = 6siu — (b) u(x, 0) = L Lnary differential equations are ioplied by the method of ‘ L separation of variables? 3yrx

* (c) a (x, 0) = 2 cos—LOei A B ( iii \ tin i2ii iii 1 1 0 <x L/2r—) (b)*(a) — — () a (x. 0)Or rdr Or I Br = Ais Ox = j 2 L/2 <x 82u 02u tin 1 ii / tin” (e) u(.r, 0) = f(s)

Ca— I*(c) —+----=O Br)Ox- Or- Br r- Or [Your answer in part (u) nras involve certain integrals that

Ott 04u Btu O2ir cia lint treed to be evaluated.]‘i’(e) ——k it — Ox

* (1) .—

c2 4. Consider

2. Consider the differential equation it Ox- subject to u(0, t) = 0, u(L, t) 0, and it(x, 0) = f(x).(fçl

+ ?4 = 0. °(a) What is the total heat energy in the rod as a function of time?

Determine the eigenvaJues A (and corresponding eigen. functions) if’ satisfies the following boundary conditions. (b) What is the flow of heat energy out of the rod at Analyze three cases (?e> 0,A = 0,3 <0). You mayan- x0? ntx = stone that the etgenvalues are reaL. n(C) What relationship should exist between parts (a) and (a) •(0) = 0and(ar)=0 (b)7

¶b) •(0) = 0 and (I) = 0 5. Evaluate (be careful if n = in)

,tyrx . ,nyrx(c) (0) 0 and P(L) = 01Ff necessaty, see Sec d’c di sin —sin——--— dx forn >0.,ti >0. tion2.4.l.) L L

dId Use the trigonotnetric identity“(d) th(0)=Oand—(L)=0

(c) (0) = Oand 1(L) = 0 5100 slob = [cos(a — B) — cos(a + b)1,dx - °(f) •() = 0 and (b) = 0 (You may assunie that A >0.) . Evaluate

dl ttsrx tttn.r(g) Ii(0) = 0 and —(L)+ (L) = 0(Ifnecessrny, see r cns—cos———dx forti > On, >0.Section 5.5.) j1 L L —

3. Consider the heat equation Use the trgononietric ideittity

it, 02n con b = 4 [cos(a + h) + cos(a — b)l. Or _k court -

subject to the houodary conditions (Be careful if a — & = 0 or a + b = 0.) 7. Consider the following boundary value problem (if neces

t,(0,t) = 0 and u(,t) = 0. sary, see Section 2.4.1): Ba dn On On

— k— with —(0 t) = 0. —(L,t) = 0. andSolve the initial value problem if the temperature is mi- — Ox’ O.r ‘ B.c

tially et)x,0) = f(x).

774 Chapter 13 Method of Separation of Variables

forms the fundamental part of the boundary value problem. We collect in the table in one place the relevant Irirmulas for the eigenvalues and eigenfunctions for the typ ical boundary conditions already discussed. You will find it helpful to understand these results because of their enormous applicability throughout this text. It is im portant to note that, in these cases, whenever A. 0 is an cigenvalue, a constant is the cigenfunction (corresponding toii 0 in cosnzr.’c/L). For closed-book exam inations. instructors might find it useful to provide Table I 3.4.0, though it may be helpful for students just to memorize all cases,

(0) = (L) Unundary b (0) = 0 dx conditions

,ar 2

a(L)=0 (L) = (L)

Elgenvalues ‘a ( T) ( L ) L a = 1,2, 3,.. a =0. 1,2,3,... a =0, 1,2, 3,...

. telex ,l,r.r - flTX 527% Eigenfunctiuias sin con —i—— sin

.‘—-—

and cos

f(x) = il,COS L

Series flu) = L rtst = con ‘

“ ,

L °=

Au = — / f(u) dx2 L ,or.a L L

Coefficients B, / f (a) sin ai.r . L = f f(r) cos —i-- dx A11=--f f(.S)COsfIfLdX L0

h2 = -f f)x)sinOiJ dx

Problems

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1. Solve the heat equation On/fit = kfi2rn/8x2, 0 < x < L, t>0,subjeclto

0 in —(0,r)=0 t>0

Oil —(L,t)=0 t>0. Ox

10 .u’<L/2 hex(a) a (x. 0) = I, x > L/2 (6) U (x, 0) = 6 + 4 cos

For this problem you may assume that no solutions of the heat equation exponentially grow in time. You may also guess appropriate orthogonality conditions for the eigen functions.

3. Solve the eigenvalue problem

subject 10

4. Explicitly at

d2)b = —A)dx

5. This problee equation foe of finite thic polar coord temperature thin.

6. Determine thin circular (a) directly

tion 12. (6) by cow

dependi

d2cb TABLE 13.4.1. Bounclany Value Problems for —s’ =—dx”

2. Solve On 02n . Ba — =k— with —(0, f) 0 fit Ox

u(x,0) = f(x).

(c) u(x,0)=—2sin’ (d) u(x,0) = —3cos 8ir d2 =

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