Application of modified Jacobi partial differential operators to partial differential equations

              Rev . Acael.Canar . Cienc ., IX (Núm . 1), 47-56 (1997)
APPLIC/\'rION OF MODIF'IED JACODI PARTI/\L DIFFEnENTIAL
OPEílATOHS TO PAflTIAL DIFFERENTI.AL EQUATIONS
M.C. MUKHRRJSE
No taj ina~ar Vidy amandir
Calcutta-700092
INDIA
l. Introduction,
Ro contly, Isanc 1.¡¡. Chon and T,W, Barrot;t [_1] hnvo solved
som'3 sccond order linear ordinary differ-:rntial equations with the
hclp of Bosscl 's ortlinary diff~ronttul op:!rators which raisc and
1
lower thc indox of Bessel 's function of the first kind. The ohjoct
or tllis pa¡mr is tu use tilo partinl di.fforential operators in
connection wi th the modificd Jacobi oolynomials, which are ro .:;;ardod
as gcncrators or Lio-alcobra, in the detcrmination of somo opcra­tional
rosults and finnlly in thc solution of thosc partj.nl diff­orvnt1nl
oquntionn which con bo fnct.orincd by mout1!J of tllu ¡;ono rators
of t!Je Lie-nlgebra ~ar modified Jacobi polynomials.
How, \ta considor
"y a o ( 1.1) ( -b:;- [ (x-a) (b-x) -óx- - y(a+b-2x) -i,y- ( + { a (b-x) - ~(x-n) JJ Xn Xn+l
b- a. -1 (-- -- - y -~-- -) X = X ,
" 'é)x n n-..L
47
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wher,.:
F n (a-n , '~ -n ,• x)
is the extended Jacobi pclynomb.ls.
I f we subs ti tute
R = "A.y [
b-a
(1.2)
b-a
L = y
"
then
(1.3)
where
(x-a)(b-x)
-1 'h ox
oox - - y(a+b-2x) o o :;;-
::
+ { a(b-x) - ~(x-a))l
(
-(n+l) fn+l (x ,Y) )
(l+a+p-n) fn(x,y)
We also note tnat
which yie lds tile followin¿; rela tion
( 1.5)
2 02
[ ( ;,:- a)(b- :-:) _Q __ + ( 2x- a-b) y-- --- --
'o/ oxa y
+{(a+l)(b-x; - (~ü)(;:-a)}()~-+2y ;gy- - (n+fju O
48
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Again
[ R ,L] = ( a +f3 ) - 2y 'o~
where
[ R ,L J = RL - LR
Also we have
A b-a -1
( 1.6) b-a
y(x-a) (b-x) L - -- y R
A
= y(a+b-2x) _oay__ - {_ a (b-x) - ~ (x-a)}
>.. o y L = ox b-a
2. Derivation of Operational Formulas from Modified Jacobi
Raising and Lc·~'8ring Operators~
we consider :l1e partial diffeNntial equDticn
(2.1) ~:a l. (z-a(b-x) ~~ - y(a+b-2x) ~~
+ t a(t-x) - f(x-a) Ju J = f(x,y)
whlch is eq ui valent to
(x- 2 ) (b-x) -1i):)xu- - y(a+b-2x) ~uy u + JL a (b-x) - F (x-:i}J u
-1
(b-o.)y
= ---- --- - - - f( x ,y)
A
49
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Hencu tlJíJ corrrJ3ponclinr, :,ystu:n of orJinary clifforential et1untions
is
clx
(2.2)
(x-a) (b-x)
solving (2.2) we get
(x-a) (x-b)y
and
a p
(2,3) u(x-a) (x-b) =
dy
-y(a+b-2x)
1 b-a - -- I
el :\
b-:i
>-
clu
-1
y f(x,y)
- {a (h-x)-f(Y.- :i) 5 u
a ~
(x-a) (x-b)
el
X f ( X' ------------) clx + c..,
(x-a) (x-b) ~
( say) .
He neo
(2.4)
-a -p
= (x-a) (x-b) ~1 (x,e1 )\
el ( z- :i) ( x-h)
-u -p
+ (x-a) (x-b) ~1 ((x-a)(x-b)y)
whor(;J \Vl is givon in (~.:3) ancl <t>1 is arlJitrary.
'i()
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Corollaries :
(1) If we take
y -1 f(x,y) = a func~ion of x only
then
(2.5) a1 rcx,y)
= P(x)
-a
(x- a)
-p a-b a-1 ~-1
(x-b) (--) J (x-a) (x-b) .
~
-ci -p
x P(x) dx + (x-s) Cx-b) ~i<Cx-a) (x-b)y)
whe re ,ii1 is gi ven in ( 2 .3) and o1 is arbitrar¡.
(ii) If we take
a-2 r-2
(x-a) (x-b)
(2x-a-b)
f(x,y) = a fur.ction of y
= Q(y) (say)
only
( 2 ,6)
-a -p a-b -2
(x-a) (x-b) (-.>,,-) J Y Q(y) dy
+
whe re <:>1 is arbi trary.
-a
( x- a)
t
( x-b) ~ ((x-a)(x-b)y)
1
llex t ,,...:; consider t he pa , tial diff,:;nr.tial equ.1 ti c:1
( 2 . 7 )
b- a -1
y ou
ox = F(x,y)
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which is equivalent to
du
dx
+ o ou
OY
A
b-3.
y F(x,y)
m~nce tlie corr~s ;Jon:Enr. systa:.1 of ordin~r:,.: difr\ir·;;nti.al equation
is
(2. 8)
dx
l
dy
= = o
du
(-_A_) Y ~r ( x,y )
b-a
Solvine (2.8) we get
and
(2.9) u
"
:: (say)
Hence
( 2. 10)
1
L (F(x,y))
where is given in (2.9) and is arbi trary .
corollaries :
(i) If y F(x ,y ) = a funr.tion of x only
,,
:: P ( x) (so.y )
52
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then
(2.11)
whe r e cf>2
(11)
then
( 2 .12) l L
l (F( :: ,Y))
L
is arbi trary.
If F(x,y) =
=
(F(x,y)) =
I J P ( x) dx + cf>2 (y)
a function of y only
Q I (y) (say)
).. I J y Q (y) dy + <?>2 ( y )
b-c..
where ,:,2 is arbi trary.
3. Application of Operational Formulas to second Order Paxtial
D1fferential Equations.
\-le consider the partial diff eren tiéi l equotion
(3.1) ( x-c:.)(b-x) cfu + (2x-a-b) y ?} u ox2 oxoy
J } au ou + LCa+i)Cb-x) - CF+l)Cx-a) 'ox + 2y 7Jy - Ca+n u
= F(x,y)
Sincc
[ b~a y-1 l] [ b~a Y{ (Y.- o. )( b- x ) "ix -yU,+b- 2x) ~.­+
a (b- x) - f(x- a)J J =
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02 02
(x-2.)(b-):) --- + (2x-a-b) y --- o:<2 óx'cy
The above rnenti cne: partial diffe renti al e quation (2.1) tak::s ti:e
following for:;i
LR u = F(x,y) .
It follows fro rn ( 2 .10) that
(3.2)
(3.3) = f(x,y) ( say)
where ~12 is give:: in ( 2 .9) and <:>2 is arbitrary.
(3 .4)
Again it follows froo (2.4) that
-a -r
u = ( x-c.) (x-b) 'V1 (x,c1 ) /
c1 (x-c.)( x-b)y
+
-a
(x-a)
-p
(x-b) + <P ((:x-a) (x-b)y)
1
~ 1 1>1
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Qn the othc rll~r. r;, if we considcr tl ; LJ cqu;1ti0n
RLu f(x,y)
whi ch is eq~i va lcnt to
(3.E) e? ci2u (:·:-a) (b-x) -;:,,__; + (2x-a-b) y ---
UX:- oxoy
+ { (a +l) (b-x) - (f +l) (x- c.)} ~~ = f(x ,y)
thcn it follcws fro~ (2.4) that
(3.E)
(3.7)
where
( 3. 8)
v1hcre
\ji
l
-a -f
LU = ( x-a) (x-b) ~1 (x,c1 )1
el = (x-o.) (x-b)y
" -a -~
+ ( x-a) (x-b) ~l ( (x-a) (x-b)y)
= F(x,y) (say)
is given in {2.3) and ~l is arbitrary.
F.e nce by ( 2 .10) it follows tha t
,¡,
2
is ~ivcn in (2.9) and ~2 is arbitrary.
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l. Chen, Issac I .E,
and
Barrett, T,W.
2. Kaufmann, B.
3, McBride,E,B.
4. Wei sne r, L.
REFERENCES
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Special Functions of Mathematical
Physics from the view point of Lie
~lgebra, J. Math. Physics,
7 (1966), . 447-457.
Qbtaining Generating Functions,
Springer Verlag, Berlin (1971).
Group- theoretic origin of certain
generating functions, Pacific J.
Math. 5 (1955), 1033-1039.
56
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