Vol. 12, December 2014 1144
Solving the Partial Differential Problems Using Maple
Chii-Huei Yu
Department of Management and Information Nan Jeon University of Science and Technology Tainan City, Taiwan, R. O. C.
ABSTRACT
This paper considers the partial differential problem of two types of multivariable functions and uses mathematical software Maple for verification. The infinite series forms of any order partial derivatives of these two types of multivariable functions can be obtained using binomial series and differentiation term by term theorem, which greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, four examples are used to demonstrate the calculations.
Keywords: Partial derivatives, infinite series forms, binomial series, differentiation term by term theorem, Maple.
1. Introduction
In calculus and engineering mathematics, the evaluation and numerical calculation of the partial derivatives of multivariable functions are important.
The Laplace equation, the wave equation, and other important physical equations involve the partial derivatives. The evaluation of the m -th order partial derivative value of a multivariable function at some point, generally, requires two procedures: the determination of the m -th order partial derivative of the function, and the substitution of the point into the m -th order partial derivative. These two procedures become increasingly complex calculations for increasing order of partial derivative, thus manual calculations become difficult. The present study considers the partial differential problem of the following two types of n -variables functions
n r i
i i n
i i i
n x a b x
x x x
f
1 1
2
1, , , )
( (1)
n r
i i i
n
i i i
n x a b x
x x x
g
1 1
2
1, , , ) exp exp
(
(2) where n is a positive integer, a,b,r,i,i are real numbers for all i 1,..,n , a,b0, and
r r b
a , exist. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions using binomial series and differentiation term by term theorem; these are the major results of this study (i.e., Theorems 1 and 2), which greatly reduce the difficulty of calculate their higher order partial derivative values. The study of partial differential problems can refer to [1-24].
The methods adopted in [1-5] are different from the methods used in this paper, and [6-24]
studied the evaluation of the partial derivatives of different types of multivariable functions using differentiation term by term theorem and complex power series method. [25] considered two differential equations whose independent variables involve the partial derivatives. [26]
discussed the distance functions whose expressions contain the partial derivatives, and [27] found the solutions of some type of partial differential equation. In this article, some examples are used to demonstrate the proposed calculations, and the manual calculations are verified using Maple.
2. Main Results
Some notations used in this paper are introduced below.
Solving the Partial Differential Problems Using Maple, Chii‐Huei Yu / 1144‐1153
Journal of Applied Research and Technology 1145
2.1 Notations 2.1.1
n
i ci c c cn
1 1 2 , where n is a positive
integer, ci are real numbers for all i 1,..,n . 2.1.2
Suppose that
t
is any real number, andm
is any positive integer. Definem t )
( t(t1)(tm1), and (t)01. 2.1.3
Suppose that n is a positive integer, ji are non- negative integers for all i 1,..,n . For the n - variables function f (x1,x2,,xn) , its i
j
times partial derivative with respect tox
i for alln
i 1,.., , forms a j1 j2 jn -th order partial derivative, denoted as
) , , , ( 1 2
1 1 2 2 2 1
j n n j
nj
jn j j
x x x x
x x
f
The followings are two important theorems used in this study.
2.2 Binomial series
0 !
) ) (
1 (
k k k
r u
k
u r , where
u, r
are realnumbers, and u 1.
2.3 Differentiation term by term theorem ([28, p230]).
For all non-negative integers
k
, if the functions Rb a
gk :( , ) satisfy the following three conditions: (i) there exists a point x0(a,b)such that
0 ( 0)
k gk x is convergent, (ii) all functions
) (x
gk are differentiable on the open interval
) ,
( b a
, and (iii)
0
) (
k gk x
dx
d is uniformly
convergent on
( b a , )
, then
0
) (
k gk x is uniformly convergent and differentiable on
( b a , )
. Moreover, its derivative
0
) (
k k x dx g
d
0
) (
k gk x
dx
d .
The following is the first major result in this study, we determine the infinite series forms of any order partial derivatives of the n -variables function (1).
2.4 Theorem 1
Suppose that n is a positive integer, a,b,r,i,i are real numbers for all i 1,..,n , a,b 0, and
r r b
a , exist. If the n -variables function
) , , ,
(x1 x2 xn f
n r i
i i n
i
i i a b x
x
1 1
satisfies that xii ,xii ,xiir exist, xi 0 for all i 1,.., n , and
b x a
n i
i i
1 .
Case A. If
b x a
n i
i i
1 , then the j1j2jn- th order partial derivative of f(x1,x2,,xn)
) , , , ( 1 2
11 2 2 2 1
j n n j
nj
jn j j
x x x x
x x
f
n
i
ji k i i i n
i i i ji
k k
r k k x
a b k a r
1 1
0
)
! ( )
(
(3)
Case.B.-If-
b x a
n i
i i
1 . (1, 2, , )
11 22 2 1
j n n j
nj
jn j j
x x x x x x
f
Solving the Partial Differential Problems Using Maple, Chii‐Huei Yu / 1144‐1153
Vol. 12, December 2014 1146
n
i
ji ir k i i i n
i i i i ji
k k
k
r k r x
b a k b r
1 1
0
)
! ( )
(
(4) Proof Case A. If
b x a
n i
i i
1
Because ) , , ,
(x1 x2 xn f
n r i
i i n
i
i i a b x
x
1 1
n r i
i i n r
i
i i x
a a b
x
1 1
1
n
i ik i k k
k n r
i
i i x
a b k a r
x
1 0
1 !
)
(
(Because
b x a
n i
i i
1
, we can use binomial series)
n
i
k i i i k k
r k x
a b k a r
1
0 !
)
(
(5)
By differentiation term by term theorem, differentiating ji-times with respect to xi (i 1,..,n) on both sides of Equation (5), we have:
the j1 j2 jn-th order partial derivative of )
, , ,
(x1 x2 xn
f , ( 1, 2, , )
11 2 2 2 1
j n n j
nj
jn j j
x x x x
x x
f
n
i
ji k i i i n
i i i ji
k k
r k k x
a b k a r
1 1
0
)
! ( )
(
Case B. If
b x a
n i
i i
1
Because ) , , ,
(x1 x2 xn f
n r i
i i n
i
i i a b x
x
1 1
n r i
i i n
i ir r i n
i
i i x
b x a
b
x
1 1 1
1
n
i ik i k k
k n r
i
ir
i i x
b a k b r
x
1 0
1 !
)
(
n
i
ir k i i i k k
r k x
b a k b r
1
0 !
)
(
(6)
Using differentiation term by term theorem, differentiating ji-times with respect toxi (i1,..,n) on both sides of Equation (6), we obtain:
) , , , ( 1 2
11 2 2 2 1
j n n j
nj
jn j j
x x x x
x x
f
n
i
ji ir k i i i n
i i i i ji
k k
r k k r x
b a k b r
1 1
0
)
! ( )
(
■ 2.5 Remark 1
If b
x a
n i
i i
1 , using ratio test ([28, p193]) yields:
n
i
ji k i i i n
i i i ji
k k
r k k x
a b k a r
1 1
0
)
! ( )
(
is uniformly convergent. Thus, we can use differentiation term by term theorem to prove Equation (3) holds. The same reason that we can employ differentiation term by term theorem to confirm Equation (4) holds.
The following is the second major result in this study, we obtain the infinite series forms of any order partial derivatives of the n -variables function (2).
2.6 Theorem 2
If the assumptions are the same as Theorem 1.
Suppose that the n -variables function
n r
i i i
n
i i i
n x a b x
x x x
g
1 1
2
1, , , ) exp exp
(
satisfies that n r
i ixi
b
a
1
exp exists, and
Solving the Partial Differential Problems Using Maple, Chii‐Huei Yu / 1144‐1153
Journal of Applied Research and Technology 1147 b
x a
n
i i i
1
exp
Case A. If
b x a
n
i i i
1
exp , then the
jn
j
j1 2 -th order partial derivative of )
, , ,
(x1 x2 xn
g ,
) , , , ( 1 2
11 2 2 2 1
j n n j
nj
jn j j
x x x x
x x
g
n
i i i i
n i
ji i i k k
r k k k x
a b k a r
1 1
0
) (
exp )
! ( )
(
(7)
Case B. If
b x a
n
i i i
1
exp , then
) , , , ( 1 2
11 2 2 2 1
j n n j
nj
jn j j
x x x x
x x
g
n i
ji i i i k
k
r k k r
b a k b r
1 0
)
! ( )
(
n
i ik i ir xi
1
) (
exp (8)
Proof Case A. If
b x a n
i i i
1
exp
Because ) , , ,
(x1 x2 xn g
n r
i i i
n
i ixi a b x
1 1
exp
exp
n r
i i i
n r
i i i x
a a b
x
1 1
exp 1
exp
n
i i i
k k
k n r
i i i k x
a b k a r
x
1 0
1
! exp )
exp (
n
i i i i
k k
r k k x
a b k a r
1 0
) (
! exp )
( (9)
By differentiation term by term theorem, differentiating ji-times with respect to xi (i1,..,n) on both sides of Equation (9), we obtain: the j1 j2 jn-th order partial derivative of g(x1,x2, ,xn) ,
) , , , ( 1 2
11 2 2 2 1
j n n j
nj
jn j j
x x x x
x x
g
n
i i i i
n i
ji i i k k
r k k k x
a b k a r
1 1
0
) (
exp )
! ( )
(
Case B.
b x a
n
i i i
1
exp
Because ) , , ,
(x1 x2 xn g
n r
i i i
n
i ixi a b x
1 1
exp
exp
n r
i i i
n
i i i
n r
i i i x
b x a r b
x
1 1 1
exp 1 exp
exp
n r
i i i
n
i i i
n r
i i i x
b x a r b
x
1 1 1
exp 1 exp
exp
n
i i i
k k
n k
i i i i
r k x
b a k x r
r b
1 0
1
! exp ) ) (
(
exp
i i i i
n i k k
k
r k r x
b a k
b r exp ( )
! ) (
1 0
(10) Using differentiation term by term theorem, differentiating
j
i-times with respect to ix
(i1,..,n) on both sides of Equation (10), we obtain:
) , , , ( 1 2
11 2 2 2 1
j n n j
nj
jn j j
x x x x
x x
g
n i
ji i i i k
k
r k k r
b a k b r
1 0
)
! ( )
(
Solving the Partial Differential Problems Using Maple, Chii‐Huei Yu / 1144‐1153
Vol. 12, December 2014 1148
n
i ik i ir xi
1
) (
exp
■ 2.7 Remark 2
The same reason as that in Remark 1, we can use differentiation term by term theorem to prove Equations (7) and (8) hold.
3. Examples
For the partial differential problem of the multivariable functions in this study, four examples are proposed. Theorems 1 and 2 are used to obtain the infinite series forms of any order partial derivatives of these functions, and to evaluate some of their higher order partial derivative values.
Additionally, Maple is used to calculate the approximations of these higher order partial derivative values to verify the manual calculations.
3.1 Example 1
Suppose that the domain of the two-variables function
5 / 11 24 13 2
/ 25 3 / 17 2 1
1(x ,x ) x x (9 2x x )
f (11)
is
2 , 9
0 , 0 )
,
(x1 x2 R2 x1 x2 x13x24
3.1.1
If 2
4 9
3 2
1 x
x . Using Case A of Theorem 1 yields:
any j1 j2-th order partial derivative of )
, ( 1 2
1 x x
f ,
) , ( 1 2
11 2 2
2 1
1 x x
x x
f
j j
j j
0 1 2
5 11
2 4 5 3 3 7 9 2
! 5 11 9
j j
k k
k k k
k
2 2 4 5 1 2 3 3 7
1 k j x k j
x (12)
Therefore, the 13-th order partial derivative value of f1(x1,x2) at
5 ,4 2
3 ,
5 ,4 2 3
16 27
13 1
x x
f
0 6 7
5 11
2 4 5 3 3 7 9 2
! 5 11
9 k k
k
k k
k
2 4 9 3 3 11
5 4 2
3
k k (13)
Next, we use Maple to verify the correctness of Equation (13).
>f1:=(x1,x2)->x1^(7/3)*x2^(5/2)*(9+2*x1^3*x2^4)
^(11/5);
>evalf(D[1$6,2$7](f1)(3/2,4/5),22);
>evalf(9^(11/5)*sum(product(11/5-j,j=0..(k-1))/k!*
(2/9)^k*product(3*k+7/3-p,p=0..5)*product(4*k+5/2- q,q=0..6)*(3/2)^(3*k-11/3)*(4/5)^(4*k-9/2),k=0..
infinity),22);
3.1.2
If 2
4 9 3 2
1 x
x . By Case B of Theorem 1, the
2
1 j
j -th order partial derivative of f1(x1,x2),
) , ( 1 2
11 2 2
2 1
1 x x
x x
f
j j
j j
0 1 2
5 11
10 4 113 15
3 134 2
9
! 5 11 2
j j
k k
k k k
k
10 2 4 113 1 2
15 3 134
1 k j x k j
x (14)