samples/texts/2487562/page_7.md

For the solvability of the system of Eqs. (3.52) and (3.53), we must have

$$\begin{array}{cccc}& \mu \mathrm{\ne }\frac{0.00977909(-75.3982\alpha \pm 75.3982\sqrt{\alpha }\sqrt{\alpha +1.11547})}{\alpha },(\alpha \mathrm{\ne }0),& & \text{(3.55)}\end{array}\backslash mu\; \backslash neq\; \backslash frac\{0.00977909(-75.3982\backslash alpha\; \backslash pm\; 75.3982\backslash sqrt\{\backslash alpha\}\backslash sqrt\{\backslash alpha+1.11547\})\}\{\backslash alpha\},\; \backslash quad\; (\backslash alpha\; \backslash neq\; 0),\; \backslash tag\{3.55\}$$

showing the existence of a value of $\mu$, different from $\lambda = \frac{1}{2\alpha} \neq \mu$, giving rise to difficulties.

(C) Since the relation (1.5) (see also (1.13)) represents an over-determined system of linear equations, if we apply the least-squares method then a solvable determinate system of linear equations (see the Eqs. (1.7)-(1.9) and (1.15)-(1.17)) can be obtained.

(D) Though the above method of least-squares solution is expected to work well enough for Fredholm integral equations of the second kind, it may give rise to non-unique solutions of integral equations of the first kind involving varieties of kernels, as illustrated by the following examples:

(a)

$$f_0^1(1+xt)\varphi (t)dt=x,0\le x\le 1.f\_0^1\; (1\; +\; xt)\; \backslash phi(t)\; dt\; =\; x,\; \backslash quad\; 0\; \backslash le\; x\; \backslash le\; 1.$$

(b)

$$(b){\int }_0^1(x+t)\varphi (t)dt=1,0\le x\le 1.(b)\; \backslash int\_\{0\}^\{1\}\; (x+t)\backslash phi(t)dt\; =\; 1,\; \backslash quad\; 0\; \backslash le\; x\; \backslash le\; 1.$$

Solution:

Using the method described in Section 1, if $\phi(x)$ is approximated by the relation (1.11), then we find that $a_{i-1}(i=1,2,\dots,N+1)$ satisfy the system of normal Eqs. (1.15) where

For example (a):

$$c_{ij}=\frac{1}{\bar{i}\bar{j}}+\frac{1}{2\bar{j}(\bar{i}+1)}+\frac{1}{2\bar{i}(\bar{j}+1)}+\frac{1}{3(\bar{i}+1)(\bar{j}+1)},(3.56)c\_\{ij\}\; =\; \backslash frac\{1\}\{\backslash bar\{i\}\backslash bar\{j\}\}\; +\; \backslash frac\{1\}\{2\backslash bar\{j\}(\backslash bar\{i\}+1)\}\; +\; \backslash frac\{1\}\{2\backslash bar\{i\}(\backslash bar\{j\}+1)\}\; +\; \backslash frac\{1\}\{3(\backslash bar\{i\}+1)(\backslash bar\{j\}+1)\},\; \backslash quad\; (3.56)$$

$$\begin{array}{cccc}& b_j=\frac{1}{2j}+\frac{1}{3(j+1)}\mathrm{.}& & \text{(3.57)}\end{array}b\_j\; =\; \backslash frac\{1\}\{2j\}\; +\; \backslash frac\{1\}\{3(j+1)\}.\; \backslash tag\{3.57\}$$

By choosing $N = 1$, we find that an approximate solution is given by

$$\begin{array}{cccc}& \varphi (t)=-6+12t\mathrm{.}& & \text{(3.58)}\end{array}\backslash phi(t)\; =\; -6\; +\; 12t.\; \backslash tag\{3.58\}$$

It is verified that this $\phi(t)$ satisfies the integral equation exactly.

Again, by choosing $N=2$, we observe the above matrix [$c_{ij}$] is singular.

For example (b):

$$c_{ij}=\frac{1}{3ij}+\frac{1}{2j(i+1)}+\frac{1}{2i(j+1)}+\frac{1}{(i+1)(j+1)},(3.59)c\_\{ij\}\; =\; \backslash frac\{1\}\{3ij\}\; +\; \backslash frac\{1\}\{2j(i+1)\}\; +\; \backslash frac\{1\}\{2i(j+1)\}\; +\; \backslash frac\{1\}\{(i+1)(j+1)\},\; \backslash quad\; (3.59)$$

$$\begin{array}{cccc}& b_j=\frac{1}{2j}+\frac{1}{(j+1)}\mathrm{.}& & \text{(3.60)}\end{array}b\_j\; =\; \backslash frac\{1\}\{2j\}\; +\; \backslash frac\{1\}\{(j+1)\}.\; \backslash tag\{3.60\}$$

By choosing $N = 1$, we find that an approximate solution is given by

$$\begin{array}{cccc}& \varphi (t)=-6+12t\mathrm{.}& & \text{(3.61)}\end{array}\backslash phi(t)\; =\; -6\; +\; 12t.\; \backslash tag\{3.61\}$$

It is verified that this $\phi(t)$ satisfies the integral equation exactly.

Again, if we choose $N=2$, we encounter the same situation giving rise to a singular matrix as obtained in example (a).

The reason for encountering such singular matrices in these examples is to be attributed to the fact that the integral equa- tions here are of the first kind, which generally produce non-unique solutions. In fact $\phi(x) = -24x + 36x^2$ is another solution of both the integral equations considered in the above examples. Though we have found in the above examples that singular systems occur for integral equations of the first kind for the special choices of the order $N$ of the polynomial solutions where exact solutions become available for $N-1$, it is not straight forward to establish the opposite fact in general.

4. Conclusion

In order to solve a special class of Fredholm integral equations of the second kind the unknown function is approximated by a polynomial and the least-squares method is used to solve the resulting over-determined system of equations. Several illustrative examples are examined in detail.

Acknowledgements

A. Chakrabarti thanks the University Grants Commission, New Delhi, India for awarding him an Emeritus Fellowship and S.C. Martha is grateful to the National Board for Higher Mathematics (NBHM), India for providing a Post Doctoral Fellowship.

References

[1] B.N. Mandal, S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Appl. Math. Comput. 190 (2007) 1707–1716.