I f I
C >
--
t
APM3I DI ~
~St·SIGNMENT 1
l r r r I
---.--lt:I: QUEs--11!1 NS Fill I:t.Yi- AN rw E_D
E ...____
R >----------+---
a1scLA1MER: l I
- THIS-orlcu,EN l~FDR-RE ERENCE-ANO-GUID~NcE-PURP ES
- DNbY.---1 DD NOT TAKE-RESPDNSIBIL!ITY~FDR ANY PLAGIARI M. MISUS .-DR- -
ACADEMIC MISCONDUCT RESUliTING FROM THEIR USE. IT IS YOUR j
RESPDNSIBILITf jD ENSU~E ~IGINAL~TY '.AN CD PL!IAN E WITH _
RELEVANT GUI ELINES DR1sTANDARD9. t ~
, PLEASE USE THIS DOCUMENT AS A GUIDE ONLY
QUESTION 1
1. Solve the following (initial)-boundary value problem, (Check your answer by substituting,
and explain all the steps clearly)
Part a
1. Problem Statement
We are asked to solve · he partial differenti.al equat ion (PDE}:
Given the conditfons •
2. Step-b:y-.Step Sol utioni
1
Ste p 1 1.: Integrate w"th respect to y
Integrate bo h sides of t he PDE ith respect to '!F
flu
2
-- =
o-x8t
j. .
2- xt.dy = 2 lJt ~ (x )
·w here F X" t )i ~ an arbitra ry. fu n.c ion of a 11d t .
Step 2:: Determin.e F ( , .} 1u si ng the third con.dition
We a re given t hat when y = 0:
If u
.
&x.·&,·
.( · , 0 t = 2- xt ,• - •
, Substitute y = 0 into our integrated equation:
2z{O)t + F (z, t) = 2zt + x - L = F (x. t) = 2:rt + z - t
So, the equation becomes:
cl2u
OxOt = 2xyt + 2xt + x - L
Step J: Integrat e with respect to L
Now, integrate with respect to t:
~= f (2x yt + 2xt + x - t)dt
Ou t2
- = xyt2 + zt2 + zl - - + G(x, y)
8z 2
where G(x, y) is an arbitrary function of x and y .
Step 4: Determine G(x, y) using the second condition
We are given that when t = 0:
au xy
ax (x,y , O) =2
Substitute t = 0 into our expression for ~u:
,,x
o + o+ o - o + G(x, y) = x: = G(x, y )
xy
=2
So, the first derivative with respect to x is:
au ,, l2x y
- = xyt- + xl + xl - - + -
2
ax 2 2
C >
--
t
APM3I DI ~
~St·SIGNMENT 1
l r r r I
---.--lt:I: QUEs--11!1 NS Fill I:t.Yi- AN rw E_D
E ...____
R >----------+---
a1scLA1MER: l I
- THIS-orlcu,EN l~FDR-RE ERENCE-ANO-GUID~NcE-PURP ES
- DNbY.---1 DD NOT TAKE-RESPDNSIBIL!ITY~FDR ANY PLAGIARI M. MISUS .-DR- -
ACADEMIC MISCONDUCT RESUliTING FROM THEIR USE. IT IS YOUR j
RESPDNSIBILITf jD ENSU~E ~IGINAL~TY '.AN CD PL!IAN E WITH _
RELEVANT GUI ELINES DR1sTANDARD9. t ~
, PLEASE USE THIS DOCUMENT AS A GUIDE ONLY
QUESTION 1
1. Solve the following (initial)-boundary value problem, (Check your answer by substituting,
and explain all the steps clearly)
Part a
1. Problem Statement
We are asked to solve · he partial differenti.al equat ion (PDE}:
Given the conditfons •
2. Step-b:y-.Step Sol utioni
1
Ste p 1 1.: Integrate w"th respect to y
Integrate bo h sides of t he PDE ith respect to '!F
flu
2
-- =
o-x8t
j. .
2- xt.dy = 2 lJt ~ (x )
·w here F X" t )i ~ an arbitra ry. fu n.c ion of a 11d t .
Step 2:: Determin.e F ( , .} 1u si ng the third con.dition
We a re given t hat when y = 0:
If u
.
&x.·&,·
.( · , 0 t = 2- xt ,• - •
, Substitute y = 0 into our integrated equation:
2z{O)t + F (z, t) = 2zt + x - L = F (x. t) = 2:rt + z - t
So, the equation becomes:
cl2u
OxOt = 2xyt + 2xt + x - L
Step J: Integrat e with respect to L
Now, integrate with respect to t:
~= f (2x yt + 2xt + x - t)dt
Ou t2
- = xyt2 + zt2 + zl - - + G(x, y)
8z 2
where G(x, y) is an arbitrary function of x and y .
Step 4: Determine G(x, y) using the second condition
We are given that when t = 0:
au xy
ax (x,y , O) =2
Substitute t = 0 into our expression for ~u:
,,x
o + o+ o - o + G(x, y) = x: = G(x, y )
xy
=2
So, the first derivative with respect to x is:
au ,, l2x y
- = xyt- + xl + xl - - + -
2
ax 2 2
