Contents
1 General Introduction 1
1.1 Background of Study . . . . . . . . . .1
1.2 Integral Equation . . . . . . . . . .......2
1.2.1 Fredholm integral equation . . .3
1.2.2 Volterra integral equation . . . . 4
1.3 Polynomials . . . . . . . . . . . . . . . . . 4
1.4 Orthogonal Polynomials . . . . . . .5
1.5 Chebyshev Polynomials . . . . . . . 6
1.5.1 Chebyshev polynomial of the first kind Tr(x) . . . . . 6
1.5.2 Chebyshev polynomial of second kind Ur(x) . . . . . 6
1.5.3 Chebyshev polynomials of third-kind Vr(x) . . . . . . 7
1.5.4 Fourth kind Chebyshev polynomial Wr(x) . . . . . . . 7
1.6 Shifted Chebyshev polynomials .7 Error Estimation . . . . . . . . . . . . . . . . 8
1.8 Purpose of Study . . . . . . . . . . . . .8
1.9 Aim and Objectives . . . . . . . . .... 9
1.10 Scope and Limitations . . . . . . . 9
2 Literature Review .......................10
2.1 Orthogonal Polynomials . . . . . .10
2.2 Chebyshev Polynomials . . . . . 11
2.3 Spectral Methods . . . . . . . . . . . 13
2.4 Volterra Integral Equations........14
2.5 Trial Solutions . . . . . . . . . . . . . . 16
2.6 Approximations . . . . . . . . . . . . 17
3 Methodology .....................19
3.1 Trial Solution Derivation Via Chebyshev polynomials . . . . . . . . . 19
3.2 Shifted polynomial . . . . . . . . . . 20
3.3 Residual solution . . . . . . . . . . . .21
4 Results and Discussions .............24
4.1 Example 1 . . . . . . . . . . . . . . . . . .24
4.2 Example 2 . . . . . . . . . . . . . . . . . 27
4.3 Example 3 . . . . . . . . . . . . . . . . . 30
4.4 Discussion . . . . . . . . . . . . . . . . . 34
5 CONCLUSION AND RECOMMENDATIONS .....................35
5.1 Conclusion . . . . . . . . . . . . . . . . .35
5.2 Recommendation . . . . . . . . . ... 35
REFERENCE . . . . . . . . . . . . . . . . . .. 36
APPENDIX . . . . . . . . . . . . . . . . . . . . .38
