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A note on the kernel approximation in a certain axially symmetric temperature problem
ht. 1. Eneng SC;. Vol. Il. pp. 345-348 0 Pergamon Press Ltd., 1979. Printed in Great Britain
A NOTE ON THE KERNEL APPROXIMATION A CERTAIN AXIALLY SYMMETRIC TEMPERATURE PROBLEM
IN
LAWRENCE 1. EBONG College of Technology, Calabar, Nigeria (Communicated by I. N. SNEDDON) Abstract-A contour integral technique and the second mean value theoremof integralcalculusare used to obtain an approximation to the kernel of the integral equation associated with a certain axially symmetric temperature problem. The method of derivation also provides the scheme by which the approximation can be improved to any desired degree of accuracy. Numerical results are obtained to demonstrate the effectiveness of the scheme. I. INTRODUCTION
recent paper on the axially symmetric temperature problem for a half-space with a crack, Par-ton [ l] reduces the problem to a Fredholm integral equation of the second kind with symmetric kernel. This equation is solved approximately by the method of kernel approximation. The approximate kernel used in that paper satisfies most of the requirements discussed in [2,3] but its choice seems rather intuitive, if not arbitrary. The object of this note is to provide a mathematical derivation of that approximate kernel and the scheme by which the approximation can be improved to any desired degree of accuracy. IN
A
disc-shaped
2. THE METHOD OF DERIVATION
The kernel of the integral equation derived in [l] can be put in the form m
my,
x) =
I
05x,
g(u, p) sin ux sin uy du,
ysl,
0
(2.1)
where g(u, p) = 1 -
u(u2+p2)-“‘2’
(2.2)
and p is some parameter. One method of approach which is that adopted in [ll is to approximate the function g(u, p) by some other function which does not only render the equation easily integrable but also preserves the essential characteristics of the original kernel. We shall take a rather different approach to derive the approximate kernel. To this end we apply a modification of Noble’s contour integral technique[4] to eqn (2.1) to obtain u(p2_
K(Y, x) =
e +
u2)-w2)
e-uy
sinh uy du, x > y, sinh ux du, x < y,
(2.3)
Let us now consider the integral Z(p, z) = 1 u(p2 - u2)-o’2)e-“’ du,
z > 0.
(2.4)
An elementary change of variable gives n/2
QP, 2) = P
sin w eepzG”wd rv. 345
(2.5)
346
LAWRENCE LEBONG
The integrand in the last equation satisfies the conditions for the application of the Second Mean Value Theorem of integral calculus[5(a)]. To generate the nth degree approximation we subdivide the interval (0, 7r/2)into n subintervals in each of which we apply the second integral mean value theorem to obtain Z(p, 2) =
$ qie 41i. I I
2
(2.6)
where 4; =
aniP
(2.7)
q”i = sin dni and
It follows from the above results that (2.8)
Expanding the exponential functions in eqns (2.4) and (2.6) and comparing coefficients of powers of pz, we obtain the following system of equations in the unknowns a.,
j=O,l,.........
(2.9)
It is obvious that eqns (2.9) cannot be satisfied exactly for all values of j, Considering the fact that the first terms in the expansions of the kernel in ascending powers of p from eqns (2.3) and (2.8) contain an even power of p we therefore seek to derive an approximate kernel by comparing only even powers of a,i in eqns (2.9) for j = O,l, . . . . . , n. That is
.
nI( j + 3/2) n”*I( j + 2)’
j=O,l ,.,..
n.
(2.10)
It is a simple matter to show that aii (i = l,2, . . . , n) are the roots of the algebraic equation
$b+“-‘=o.
(2.11)
bo= 1.
where the coefficients bi are derivable from Newton’s formula[6] k-l
2
b&,
+ kbk = 0
i=O
and
s&Z.:. i=l
The solution of eqn (2.11) involves enormous computational labour especially for large values of n. We shall, therefore, explore other means of obtaining its roots without actually solving the equation. It is easy to deduce from eqn (2.7) that the roots of eqn (2.11) can be represented in the form q$=COS*~(i+c),
1,2 ,...,
ft,
(2.12)
A note on the kernel approximation in a certain axially symmetric temperature problem
341
where the constant c is yet unknown. That this representation satisfies eqn (2.10) is easy to establish. We have i=1,2 ,...,
~a3=&cos2’,(i+c),
n.
(2.13)
But V(b)1 cos*jx
Ui + l/2)
=
?r”T(j
+
Ak cos2kx
1)
, I
where Ak =
UTi+ Ul* r(j+k+
l)r(j-k+
1)’
Using this result in eqn (2.13) and summing over i from 1 to n, we obtain essentially the result expressed in eqn (2.10). Referring again to the first terms in the expansions of eqns (2.3) and (2.8) we find that, for n=l
a:,=co2 ?r(1+ c) = 1 and it follows that c=---, 3lr 4 U$ = cos* &(4i - 3),
(2.14) i=
1,2 ,...,
n.
(2.15)
This completes the derivation of the approximate kernel. For n = 1 the approximation reduces to that given in [l]. It should be observed that eqn (2.8) can also be obtained directly by replacing the function g(s p) in eqn (2.1) with the approximate form (2.16) and evaluating the infinite integral using [71. This is precisely the approach adopted in [l] for n = 1. Following from the classical definition of an integral the approximations given in eqns (2.8) and (2.16) should be expected to approach their exact values as n gets very large. In Table 1 we Table I X
G(x)
G,(x)
G,(x)
0.0
1.0000 0.5528 0.2929 0.1679 0.1056 0.0715 0.0513 0.0385 0.0299 0.0238 0.0194
1.oooo
0.6667 0.3333 0.1818 0.1111 0.0741 0.0526 0.0392 0.0303 0.0241 0.0194
1.0000 0.5556 0.2929 0.1679 0.1056 0.0715 0.0513 0.0385 0.0299 0.0238 0.0194
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
348
LAWRENCE I. EBONG
compare the functions G(x) = 1 - x( 1 + x’)- ““’
and G.(x)=~$
#I
aii(x*+ aii)-’
for n = 1 and 3. There is indeed a remarkable agreement between the exact and the approximate functions for n = 3. 3. CONCLUSION
A contour integral technique and the second mean value theorem of integral calculus have been used to generate an approximate kernel of nth degree accuracy for the symmetric kernel of the integral equation considered by Parton. This method of derivation provides a mathematical basis for the choice of the approximating function (2.16) and can be used to establish the convergence of the approximating solution sequence to the exact solution. REFERENCES
[I] V. Z. PARTON, J. Appl. Math. Mech. 36, 104 (1972). [2] G. F. CARRIER, J. Appl. Phys. 10, 1769 (1959). 131 G. F. CARRIER, M. KROOK and C. E. PEARSON, Functions of a Complex Variable, pp.393-398. McGraw-Hill. New York (1%6). [4] B. NOBLE, In IEEE Trans. Antennas and Propagation AP-7, 337 (1959). [S] E. T. WHITTAKER and G. N. WATSON, A Course of Modem Analysis, (a) 65. (b) 191. Cambridge University Press [6] :f !.?JSPENSKY, Theory of Equations, pp. 260-261. McGraw-Hill, New York (1948). [7] A. ERDELYI, W. MAGNUS, F.OBERHETTINGERand F. G. TRICOMI, Tables of Integral Transfons. McGraw-Hill. New York (1954) (Received 9 September 1977)
Vol. 1%P. 78.
A NOTE ON THE KERNEL APPROXIMATION A CERTAIN AXIALLY SYMMETRIC TEMPERATURE PROBLEM
IN
LAWRENCE 1. EBONG College of Technology, Calabar, Nigeria (Communicated by I. N. SNEDDON) Abstract-A contour integral technique and the second mean value theoremof integralcalculusare used to obtain an approximation to the kernel of the integral equation associated with a certain axially symmetric temperature problem. The method of derivation also provides the scheme by which the approximation can be improved to any desired degree of accuracy. Numerical results are obtained to demonstrate the effectiveness of the scheme. I. INTRODUCTION
recent paper on the axially symmetric temperature problem for a half-space with a crack, Par-ton [ l] reduces the problem to a Fredholm integral equation of the second kind with symmetric kernel. This equation is solved approximately by the method of kernel approximation. The approximate kernel used in that paper satisfies most of the requirements discussed in [2,3] but its choice seems rather intuitive, if not arbitrary. The object of this note is to provide a mathematical derivation of that approximate kernel and the scheme by which the approximation can be improved to any desired degree of accuracy. IN
A
disc-shaped
2. THE METHOD OF DERIVATION
The kernel of the integral equation derived in [l] can be put in the form m
my,
x) =
I
05x,
g(u, p) sin ux sin uy du,
ysl,
0
(2.1)
where g(u, p) = 1 -
u(u2+p2)-“‘2’
(2.2)
and p is some parameter. One method of approach which is that adopted in [ll is to approximate the function g(u, p) by some other function which does not only render the equation easily integrable but also preserves the essential characteristics of the original kernel. We shall take a rather different approach to derive the approximate kernel. To this end we apply a modification of Noble’s contour integral technique[4] to eqn (2.1) to obtain u(p2_
K(Y, x) =
e +
u2)-w2)
e-uy
sinh uy du, x > y, sinh ux du, x < y,
(2.3)
Let us now consider the integral Z(p, z) = 1 u(p2 - u2)-o’2)e-“’ du,
z > 0.
(2.4)
An elementary change of variable gives n/2
QP, 2) = P
sin w eepzG”wd rv. 345
(2.5)
346
LAWRENCE LEBONG
The integrand in the last equation satisfies the conditions for the application of the Second Mean Value Theorem of integral calculus[5(a)]. To generate the nth degree approximation we subdivide the interval (0, 7r/2)into n subintervals in each of which we apply the second integral mean value theorem to obtain Z(p, 2) =
$ qie 41i. I I
2
(2.6)
where 4; =
aniP
(2.7)
q”i = sin dni and
It follows from the above results that (2.8)
Expanding the exponential functions in eqns (2.4) and (2.6) and comparing coefficients of powers of pz, we obtain the following system of equations in the unknowns a.,
j=O,l,.........
(2.9)
It is obvious that eqns (2.9) cannot be satisfied exactly for all values of j, Considering the fact that the first terms in the expansions of the kernel in ascending powers of p from eqns (2.3) and (2.8) contain an even power of p we therefore seek to derive an approximate kernel by comparing only even powers of a,i in eqns (2.9) for j = O,l, . . . . . , n. That is
.
nI( j + 3/2) n”*I( j + 2)’
j=O,l ,.,..
n.
(2.10)
It is a simple matter to show that aii (i = l,2, . . . , n) are the roots of the algebraic equation
$b+“-‘=o.
(2.11)
bo= 1.
where the coefficients bi are derivable from Newton’s formula[6] k-l
2
b&,
+ kbk = 0
i=O
and
s&Z.:. i=l
The solution of eqn (2.11) involves enormous computational labour especially for large values of n. We shall, therefore, explore other means of obtaining its roots without actually solving the equation. It is easy to deduce from eqn (2.7) that the roots of eqn (2.11) can be represented in the form q$=COS*~(i+c),
1,2 ,...,
ft,
(2.12)
A note on the kernel approximation in a certain axially symmetric temperature problem
341
where the constant c is yet unknown. That this representation satisfies eqn (2.10) is easy to establish. We have i=1,2 ,...,
~a3=&cos2’,(i+c),
n.
(2.13)
But V(b)1 cos*jx
Ui + l/2)
=
?r”T(j
+
Ak cos2kx
1)
, I
where Ak =
UTi+ Ul* r(j+k+
l)r(j-k+
1)’
Using this result in eqn (2.13) and summing over i from 1 to n, we obtain essentially the result expressed in eqn (2.10). Referring again to the first terms in the expansions of eqns (2.3) and (2.8) we find that, for n=l
a:,=co2 ?r(1+ c) = 1 and it follows that c=---, 3lr 4 U$ = cos* &(4i - 3),
(2.14) i=
1,2 ,...,
n.
(2.15)
This completes the derivation of the approximate kernel. For n = 1 the approximation reduces to that given in [l]. It should be observed that eqn (2.8) can also be obtained directly by replacing the function g(s p) in eqn (2.1) with the approximate form (2.16) and evaluating the infinite integral using [71. This is precisely the approach adopted in [l] for n = 1. Following from the classical definition of an integral the approximations given in eqns (2.8) and (2.16) should be expected to approach their exact values as n gets very large. In Table 1 we Table I X
G(x)
G,(x)
G,(x)
0.0
1.0000 0.5528 0.2929 0.1679 0.1056 0.0715 0.0513 0.0385 0.0299 0.0238 0.0194
1.oooo
0.6667 0.3333 0.1818 0.1111 0.0741 0.0526 0.0392 0.0303 0.0241 0.0194
1.0000 0.5556 0.2929 0.1679 0.1056 0.0715 0.0513 0.0385 0.0299 0.0238 0.0194
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
348
LAWRENCE I. EBONG
compare the functions G(x) = 1 - x( 1 + x’)- ““’
and G.(x)=~$
#I
aii(x*+ aii)-’
for n = 1 and 3. There is indeed a remarkable agreement between the exact and the approximate functions for n = 3. 3. CONCLUSION
A contour integral technique and the second mean value theorem of integral calculus have been used to generate an approximate kernel of nth degree accuracy for the symmetric kernel of the integral equation considered by Parton. This method of derivation provides a mathematical basis for the choice of the approximating function (2.16) and can be used to establish the convergence of the approximating solution sequence to the exact solution. REFERENCES
[I] V. Z. PARTON, J. Appl. Math. Mech. 36, 104 (1972). [2] G. F. CARRIER, J. Appl. Phys. 10, 1769 (1959). 131 G. F. CARRIER, M. KROOK and C. E. PEARSON, Functions of a Complex Variable, pp.393-398. McGraw-Hill. New York (1%6). [4] B. NOBLE, In IEEE Trans. Antennas and Propagation AP-7, 337 (1959). [S] E. T. WHITTAKER and G. N. WATSON, A Course of Modem Analysis, (a) 65. (b) 191. Cambridge University Press [6] :f !.?JSPENSKY, Theory of Equations, pp. 260-261. McGraw-Hill, New York (1948). [7] A. ERDELYI, W. MAGNUS, F.OBERHETTINGERand F. G. TRICOMI, Tables of Integral Transfons. McGraw-Hill. New York (1954) (Received 9 September 1977)
Vol. 1%P. 78.