Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation
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This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces Open image in new window and Open image in new window .
KeywordsBanach Space Fixed Point Theorem Distinct Point Dense Subset Lipschitz Condition
Section 2 shows that operator Open image in new window satisfies the hypothesis of the Banach fixed point theorem and thus the sequence Open image in new window converges to the solution Open image in new window of (1.1) for any Open image in new window However, such a sequence cannot be determined in an explicit way. The method we present consists of replacing the first element of the convergent sequence, Open image in new window by the new easy to calculate function Open image in new window and in such a way that the error Open image in new window is small enough. By repeating the same process for the function Open image in new window and so on, we obtain a sequence Open image in new window that approximates the solution Open image in new window of (1.1) in the uniform sense. To obtain such sequence, we will make use of some biorthogonal systems, the usual Schauder bases for the spaces Open image in new window and Open image in new window , as well as their properties. These questions are also reviewed in Section 2. In Section 3 we define the sequence Open image in new window described above and we study the error Open image in new window . Finally, in Section 4 we apply the method to two examples.
Volterra integro-differential equations are usually difficult to solve in an analytical way. Many authors have paid attention to their study and numerical treatment (see for instance [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] for the classical methods and recent results). Among the main advantages of our numerical method as opposed to the classical ones, such as collocation or quadrature, we can point out that it is not necessary to solve algebraic equation systems; furthermore, the integrals involved are immediate and therefore we do not have to require any quadrature method to calculate them. Let us point out that our method clearly applies to the case where the involved functions are defined in Open image in new window , although we have chosen the unit interval for the sake of simplicity. Schauder bases have been used in order to solve numerically some differential and integral problems (see [1, 16, 17, 18, 19, 20]).
We first show that operator Open image in new window also satisfies a suitable Lipschitz condition. This result is proven by using an inductive argument. The proof is similar to that of the linear case (see [1, Lemma Open image in new window ]).
where Open image in new window
3. A Method for Approximating the Solution
We now turn to the main purpose of this paper, that is, to approximate the unique fixed point of the nonlinear operator Open image in new window given by (1.3), with the adequate conditions. We then define the approximating sequence described in the Introduction.
The next result is used in order to establish the fact that the sequence defined in Theorem 3.1 approximates the solution of the nonlinear Volterra integro-differential equation, as well as giving an upper bond of the error committed.
Finally, since the sequence Open image in new window is bounded, Open image in new window also is. Similarly, one proves that Open image in new window is bounded (sequences Open image in new window and Open image in new window are bounded and Open image in new window and Open image in new window are Lipschitz at their second variables) and Open image in new window is bounded (sequences Open image in new window and Open image in new window are bounded and Open image in new window , Open image in new window and Open image in new window are Lipschitz at the third variables).
We have chosen the Schauder bases above for simplicity in the exposition, although our numerical method also works by considering fundamental biorthogonal systems in Open image in new window and Open image in new window .
4. Numerical Examples
The behaviour of the numerical method introduced above will be illustrated with the following two examples.
has exact solution Open image in new window
whose exact solution is Open image in new window
Absolute errors for Example 4.1.
Absolute errors for Example 4.2.
This research is partially supported by M.E.C. (Spain) and FEDER, project MTM2006-12533, and by Junta de Andaluca Grant FQM359.
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