A Numerical Method for Solving Linear Integral

Equations

Realizatori: Onichi Mihaela-PetruțaTănase Alexandru

Dogaru Adrian-NicușorCoordonator: Sporis Ligia-Adriana

Description of the Method

Consider N[y] = 0

Where:- N is an operator;- y(x) is unknown function - x the independent

variable. Let - y0(x) denote an initial guess of the exact solution y(x).

- H ^ 0 an auxiliary parameter;

- H(x) ^ 0 an auxiliary function;

- L an auxiliary linear operator with the property L[r(x)\ = 0 when r(x) — 0. Then using q e [0,1] as an embedding parameter, we construct such a homotopy

(1-q)L[ø(x;q)- (2)

Enforcing the homotopy (2) to be zero, i.e.,

we have the so-called zero-order deformation equation(3)

When q = 0, the zero-order deformation equation (3) becomes(4)

and when q = 1, since h 0 and H(x) 0, the zero-order deformation equation (3) is equivalent to

(x; 1) = y(x). (5)

Thus, according to (4) and (5), as the embedding parameter q increases from 0 to 1, q varies continuously from the initial approximation (x) to the exact solution y{x). Such a kind of continuous variation is called defor mation in homotopy.

By Taylor’s theorem, can be expanded in a power series of q as follows

(6) Where:

If the initial guess (x), the auxiliary linear parameter L, the nonzero auxiliary parameter h, and the auxiliary function H (x) are properly chosen so that the power series (6) of (x;q) converges at q=1.Then, we have under these assumptions the solution series

(8) 

For brevity, define the vector

,…,. (9)

According to the definition (7), the governing equation of ym(x) can be derived from the zero-order deformation equation (3).

Differentiating the zero- order deformatione quation(3)m times with respective to q and then

dividing by m! and finally setting q= 0, we have the so-called mth-order deformation equation

(x)), (10)(0)=0, 

Where

 Note that the high-order deformation equation (10) is governing by the linear operator L, and the term Rm((x)) can be expressed simply by (11) for any nonlinear operator N.

Therefore, ym{x) can be easily gained, especially by means of computational software such as MATLAB. If tends uniformly to a limit as n, then this limit is the required solution.

Linear Integral Equations of the First Kind

Consider the linear integral equationg(x) (12)

where the upper limit may be either variable or fixed, A is a complex number, the kernel K(x, t) and g(x) are known functions, whereas y is to be determined.

Let

we can obtain from (11) that

The mth-order deformation equation (10) reduces to

(t)dt (13)

Alegand Ly = y ca operator liniar auxiliar , ca o aproximare de ordin zero catre functia potrivita y(X), solutia (x) = g(x),, este luata, parametrul h diferit de zero si functia auxiliara H(x)=1 . Aceasta este substituita in 13 pentru a obtine o simpla formula de repetare ym(x)

x), m = 1,2,

By considering the notations in (1), we can verify that

The solution y(x) of (12), becomes

Theorem 1 As long as the series (8) convergence, where ym(x) is governed by Eq.(13), it must be the exact solution of the integral equation (12).Proof. If the series (8) converges, we can writeS(x)  and it holds that

We can verify that

which gives us, according to (16), Furthermore, using (17) and the

definition of the linear operator L, we have

In this line, we can obtain that

(17)

which gives, since h and H(x)0, that

Substituting m-1(m-1(x)) into the above expression and simplifying it, we have

From (19) and (18), we have g(x) = (x,t)S(t)dt, and so, S'(x) must be the exact

solution of Eq. (12). ■

Conclusion

The proposed method is a powerful procedure for solving linear integral equa tions. The examples analyzed illustrate the ability and

reliability of the method presented in this paper and reveals that this one is very simple and effective. The obtained solutions, in

comparison with exact solutions admit a remarkable accuracy. Results indicate that the convergence rate is very fast, and lower

approximations can achieve high accuracy.