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