Numerical solving initial value problem for Fredholm type linear integro-differential equation system
Introduction
Integro-differential equations (IDE) arise in many branches of science, for example in control theory and financial mathematics. IDEs are important, but they are hard to solve even numerically, so the progress on how to solve them is slow. IDEs are equations of the form where the unknown function appears under the sign of integration and they also contain the derivatives of the unknown function.
The books edited by Green [14] and Kanwal [21] contain many different methods to solve the integral equations analytically. Numerical methods also take an important place in solving the integral equations (see [1], [12], [15], [42]).
In [27], Legendre wavelets on the interval [0,1) are introduced which reduced the solution of linear Fredholm and Volterra integral equation of the second kind to the solution of linear algebraic equations. Implicit and explicit forms of the Pouzet Volterra Runge–Kutta methods to obtain the numerical solution of a system of Volterra integral equation have been introduced in [29].
Chebyshev series solutions of linear integral and integro-differential equations and their systems have been considered in [3], [4], [38].
Compact finite difference method for second- and first-order IDEs as well as for systems of IDEs has been considered by Zhao and Corless [40].
Two methods for the numerical solution of a Fredholm IDE modelling neural networks have been proposed in [19]. The first approach is based on approximating the integral term by Gauss–Hermite or Gauss Laguerre quadrature formula with a linear interpolation to compute the approximation to the solution between the grid points in time. The second approach is based on representing the solutions in a linear combination of Lagrange's fundamental polynomials with coefficients depending on time.
Tau method to find numerical solutions of the Fredholm, Volterra and Fredholm–Volterra IDEs with standard, Chebyshev and Legendre bases is considered in [16], [17], [18], [32], [37].
A mechanization of solving Fredholm–Volterra IDEs has been proposed in [41]. The bases of mathematical mechanization are algorithm establishing and programming techniques. The procedure presented in this work provides Taylor polynomial solutions of the IDEs. Also, Taylor polynomial solutions of high-order nonlinear Fredholm–Volterra IDEs have been considered in [28].
In paper [22], [39], a matrix method called the Taylor collocation method is presented for a numerical solution of the linear IDEs by a truncated Taylor series. Using the Taylor collocation points, this method transforms the integro-differential equation into a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients.
Wavelet–Galerkin method for IDEs is presented in [2]. Rationalized Haar functions method for IDEs is considered in [24], [25].
Hybrid Legendre and block-pulse functions on the interval [0,1) to solve a linear IDE system is considered in [26]. Tau method for the numerical solution of systems of IDEs is considered in [33].
Adomian decomposition method for solving a system of integral equations is considered in the literature, for example [5], [6], [7], [9], [43], [44]. The initial value problem for an IDE has been solved in [43].
In papers [30], [31], [34], [35], [36] methods for the numerical solution of integral and IDEs have been proposed. The suggested methods are based on representing the solutions in the linear combination of Lagrange's fundamental polynomials and on approximating the integral term by the Clenshaw–Curtis quadrature formula.
In this paper, we introduce numerical expansion methods for solving a system of linear IDEs by interpolation and Clenshaw–Curtis quadrature rules (cf. [10], [11], [12], [13], [30]). Chebyshev polynomials are important in approximation theory, and Gauss quadrature appears in the theory of numerical integration. This method transforms the integral system into a matrix equation with the help of interpolation points. They are easily acquired on using the transformed matrix equation, which corresponds to the system of linear algebraic equations. As a result, the Lagrange interpolation solution is obtained.
Our focus here is on the system of Fredholm type IDEs. The basic ideas of the previous works are developed and applied to a system of high-order linear Fredholm type IDEswith the initial conditionswhere ; , and are known functions having th derivatives on an interval , , and , and are appropriate constants, denote the express of the th derivatives of the unknown functions , respectively.
We suppose that systems (1.1), (1.2) have a unique solution. However, the necessary and sufficient for a unique solution for Eqs. (1.1), (1.2) could be found in [1], [8], [11], [14], [20], [21], [23].
Section snippets
Clenshaw–Curtis quadrature
Clenshaw–Curtis quadrature rule is represented by means of the relationwhere the double prime on the summation symbol here and elsewhere indicates that the terms with suffixes and to be halved, and the points are (see [10], [11], [12], [30], [34], [35]) Chebyshev collocation points
Cauchy formula
Theorem 1 If and are continuous in x and y on the closed interval , and and are continuous, then
General technique by using interpolation
We know that the Lagrange interpolation in points , for the function iswhereTherefore we havewhere Following the previous subsection, we can write
Now, from Eq. (2.12) we can write
Method of solution for the system of Fredholm type integro-differential equations
Let has the Lagrange expansionsNext, we have
Numerical experimentation
In order to show the numerical efficiency, we consider three problems. Example 2 Consider the following initial value problem of linear Fredholm type integro-differential equations with the exact solutions (see, [33]): and .and for approximate the solution in the form (4.8).
We take the interpolation points as , and . We then have
Conclusions
- 1.
The presented method gives the solution at the points , or , , .
- 2.
The suggested methods can be applied to a system of Fredholm type integro-differential equations (see Examples 1–3).
- 3.
The suggested methods can be applied to a system of Volterra type integro-differential equations.
- 4.
Comparison of the results obtained from this method with the Tau method (see Examples 1 and 2) and rationalized Haar functions method (see Example 3) indicate that the
References (44)
- et al.
Wavelet-Galerkin method for integro-differential equations
Appl. Numer. Math.
(2000) Chebyshev polynomial solutions of systems of linear integral equations
Appl. Math. Comput.
(2004)- et al.
Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations
J. Franklin Inst.
(2005) - et al.
On the decomposition method for system of linear equations and system of linear Volterra integral equations
Appl. Math. Comput.
(2004) - et al.
The decomposition method applied to systems of Fredholm integral equations of the second kind
Appl. Math. Comput.
(2004) - et al.
Linear Volterra integro-differential equation and Schauder bases
Appl. Math. Comput.
(2004) - et al.
Solution of a system of Volterra integral equations of the first kind by Adomian method
Appl. Math. Comput.
(2003) - et al.
Numerical solution of a class of integro-differential equations by the Tau method with an error estimation
Appl. Math. Comput.
(2003) - et al.
Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases
J. Appl. Math. Modell.
(2003) - et al.
Numerical solution of a Fredholm integro-differential equation modelling neural networks
Appl. Numer. Math.
(2008)
Boundary value problems for second order integro-differential equations of Fredholm type
J. Comput. Appl. Math.
Solving linear integro-differential equations system by using rationalized Haar functions method
Appl. Math. Comput.
Taylor polynomial solution of high-order nonlinear Volterra–Fredholm integro-differential equations
Appl. Math. Comput.
Using Runge–Kutta method for numerical solution of the system of Volterra integral equation
Appl. Math. Comput.
Numerical expansion methods for solving integral equations by interpolation and Gauss quadrature rules
Appl. Math. Comput.
Numerical solution of integral equations by using combination of spline-collocation method and Lagrange interpolation
Appl. Math. Comput.
Numerical solution of Volterra linear integro-differential equations by the Tau method with the Chebyshev and Lejandre bases
Appl. Math. Comput.
Numerical solution of the system of Fredholm integro-differential equations by the Tau method
Appl. Math. Comput.
Numerical solution of special type of integro-differential equations
Appl. Math. Comput.
Lagrange interpolation to compute the numerical solutions of differential, integral and integro-differential equations
Appl. Math. Comput.
Numerical solution of functional differential, integral and integro-differential equations
Appl. Math. Comput.
Numerical solution of the general form linear Fredholm–Volterra integro-differential equations by the Tau method with an error estimation
Appl. Math. Comput.
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