Numerical solutions of the integral equations of the first kind

doi:10.1016/S0096-3003(02)00497-6

Applied Mathematics and Computation

Volume 145, Issues 2–3, 25 December 2003, Pages 413-420

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Abstract

We present new two numerical methods for the solution of non-singular integral of the first kind, which they are simple to use. Numerical comparisons indicate that they achieves comparable accuracy. Numerical results are illustrated by different examples.

Introduction

We shall consider the numerical solution of the non-singular Volterra integral equations of the first kind $∫_{−1}^{x} k(x,t)y(t) d t=g(x), −1⩽x⩽X$ where g(x) and k(x,t) are known function, y(t) is the unknown function to be determined and X is some large constant. Equations of this form are in general ill-posed, for a given k, g; (1.1) may have no solution, while if a solution exists the response ratio ∥∂y∥/∥∂g∥ to small perturbations in g may be arbitrary large. We shall consider here the use of Chebyshev interpolation. Baker [1], [4] studied the numerical treatment of integral equations, Brunner [2] investigated the solution of Volterra integral equations of the first kind by piecewise polynomials. Babolian and Delves [3] discuss an augmented Galerkin method for first kind Fredholm equations, Holyhead and Mckee [5] investigate Taylor, Multistep method for solving linear Volterra integral equations of the first kind. Lewis [6] studied the numerical solution of Fredholm integral equations of the first kind.

Section snippets

Method one

Consider the Chebyshev interpolation of k(x,t)y(t) w.r.t. t only $k(x,t)y(t)≈∑_{j=0}^{n} k(x,t_{j})y_{j} l_{j,n} (t)$ where $l_{j,n} (t)= 2δ_{j} n ∑ k=0 n^{″} T_{k} (t) cos kj π n$ $δ_{j} = 0.5, j=0,n 1, 0<j<n$ T_k(x) is the Chebyshev polynomial of the first kind and ∑^″ denotes a sum with first and last term halved.

Integrating both side of relation (2.1) from −1 to x $∫_{−1}^{x} k(x,t)y(t) d t≈∑_{j=0}^{n} R(x,t_{j})y_{j}$ where $R(x,t_{j})= 2δ_{j} k(x,t_{j}) n ∑ k=0 n^{″} I_{k} (x) cos kj π n$ $I_{k} (x)=∫_{−1}^{x} T_{k} (s) d s= T_{1} (x)+1, k=0 T_{2} (x)−1 4, k=1 T_{k+1} (x) 2(k+1) − T_{k−1} (x) 2(k−1) + (−1)^{k+1} (k^{2} −1), k⩾2$ Let x by z_i=cos((i−0.45)π/n), i

Method two

We shall consider the Chebyshev interpolation for y(x) only $y(x)≈∑_{j=0}^{n} y_{j} l_{j,n} (x)$ Substitute from (3.1) in (1.1), we get $∑_{j=0}^{n} R(x,t_{j})y(t_{j})=g(x)$ where $R(x,t_{j})= 2δ_{j} n ∑ k=0 n^{′′} A_{k} (x) cos kj π n (using FFT)$ $A_{k} (x)=∫_{−1}^{x} k(x,t)T_{k} (t) d t$ The values of integral A_k(x) may be computed analytically or numerically as follows: $A_{k} (x)=∑_{r=0}^{N} k(x,t_{r})b_{r,N} (x) cos kr π N (using FFT)$ Hence, integral equations of the first kind (1.1) are approximated by $∑_{j=0}^{n} R(z_{i},t_{j})y_{j} =g(z_{i}), i=0(1)n$ where $R(z_{i},t_{j})= 2δ_{j} n ∑ k=0 n^{″} A_{k} (z_{i}) cos kj π n$ $A_{k} (z_{i})=∑_{r=0}^{N} k(z_{i},t_{r})b_{r,N} (z_{i}) cos$

Numerical examples

We compare these methods on several examples two have also been considered by Lewis (1975) and Delves (1979) compared the performance of several versions of both the regularization and eigenfunction expansion algorithms for first kind.

Conclusion

1.
We have described new numerical methods for the solution of the first kind Volterra integral equations. The methods are tested by different examples.
2.
The computed errors σ_n in these tables are defined to be $σ_{n} = ∑^{n}_{j=0} e^{2}_{n} (x_{j}) n^{1/2} ≈ ∫^{1}_{−1} e^{2}_{n} (t) d t^{1/2}$ where x_jcos(jπ/n), j=0(1)n and e_n=y_n−y_exact.
3.
The very rapid convergence obtained shows that the two methods are excellent methods to treat Volterra integral equation of the first kind.
4.
In case Fredholm integral equation of the first kind, the use of Method 2

References (6)

C.T.H. Baker
The Numerical Treatment of Integral Equations
(1977)
H. Brunner
The solution of Volterra integral equations of the first kind by piecewise polynomials
J. Inst. Math. Appl.
(1973)
E. Babolian et al.
An augmented Galerkin method for first kind Fredholm equations
J. Inst. Math. Appl.
(1979)

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