Laguerre approach for solving system of linear Fredholm integro-differential equations

A numerical scheme has been developed for solving the system of linear Fredholm integro-differential equations subject to the mixed conditions using Laguerre polynomials. Using collocation method, the system of Fredholm integro-differential equations has been transformed to the system of linear equations in unknown Laguerre coefficients, which leads to the solution in terms of Laguerre polynomials. Moreover, the accuracy and applicability of the scheme have been compared with Tau method and Adomian decomposition method that reveals the proposed scheme to be more efficient.


Introduction
There are many branches of science, such as control theory and financial mathematics, which leads to integro-differential equations (IDEs). In modern mathematics, IDEs mostly occur in many applied areas including engineering, physics and biology [1][2][3][4][5][6]. The resolution of many problems in physics and engineering leads to differential and integral equations in bounded or unbounded domains. For example, problems occur in coastal hydrodynamics and in meteorology. The integrals appear in many physical contexts, containing the product of orthogonal polynomials or special functions. For example, the wave functions of the hydrogen as well as 2-, 3-and, in general, n-dimensional harmonic oscillator encompassing Laguerre polynomials and the evaluation of integrals having the product of these polynomials are essential [7].
In the fields of applied mathematics and scientific computing, spectral methods [8][9][10] became popular among researchers as a robust numerical tool. The remarkable results are obtained, using the spectral methods, to solve the problems [11][12][13] in different fields of natural sciences. Moreover, system of IDEs found in the field of science and engineering, such as nano-hydrodynamics [14], glass-forming process [15], dropwise condensation [16], wind ripples in the desert [17], modeling the competition between tumor cell and the immune system [18] and examining the noise term phenomenon [19,20]. Since analytical solutions of such type of problems are hard to determine, therefore the numerical methods are required. Many researchers presented numerical methods for system of IDEs, for instance the Tau method [21], Fibonacci matrix method [22], Bessel matrix method [23,24], Adomian decomposition method (ADM) [25], modified decomposition method [26], Galerkin methods with hybrid functions [27][28][29][30]38], differential transform method [31] and the block pulse functions method [32].
The main objective of this paper is to study the concept of the system of IDEs and manipulate the Laguerre matrix method for solving the system of linear Fredholm IDEs.
The following system of linear Fredholm IDEs has been considered where u q ðsÞ is the unknown and q k p;q ; K p;q ðs; tÞ; f p ðsÞ are the known functions defined in the interval [a, b], the kernel function K p;q ðs; tÞ can be expanded using Maclaurin series and also a k p;q ; b k p;q ; l k;p are real constants. Taking u p ðsÞ to be the approximate solution of Eq. (1) in terms of truncated Laguerre series yields where a p;n are the unknown Laguerre coefficients, to be determined for n ¼ 0; 1; 2; . . .; N, and L n ðsÞ be the Laguerre polynomial defined by ðÀ1Þ k k! n n À k s k ; 0 s b\1:

Numerical examples
Following examples have been considered to examine the reliability and efficiency of the proposed technique.
Solving the system of equations for N ¼ 3 by following the procedure stated above yields the approximate solutions u 1;3 ðsÞ ¼ 3s 2 þ 1 and u 2;3 ðsÞ ¼ s 3 þ 2s À 1 which are exactly the same as the analytical one. Table 1 shows the numerical results obtained by the proposed technique and their comparison with Tau method [21], whereas Figs. 1 and 2 depict the absolute errors e 1;3 and e 2;3 at N ¼ 3 for Example 1.

Example 3 Consider the system of Fredholm IDEs, as
subject to the following initial conditions The analytical solution is u 1 ðsÞ ¼ s 2 and u 2 ðsÞ ¼ s.
Solving the system of equations for N ¼ 2 by following the procedure stated above yields the approximate solutions u 1;2 ðsÞ ¼ s 2 À 2s þ 3 and u 2;2 ðsÞ ¼ À s 2 þ s þ 1 which are exactly the same as the analytical one. Numerical results obtained by the proposed technique are shown in Table 4.
subject to the following initial conditions The analytical solution is u 1 ðsÞ ¼ sinðsÞ and u 2 ðsÞ ¼ cosðsÞ.
Comparison of exact and proposed solutions is shown in Figs. 3 and 4 for N ¼ 3; 4 and 5, respectively.

Conclusion
In this paper, Laguerre operational matrix approach has been manipulated to solve the system of linear Fredholm IDEs. The scheme converted the system of IDEs, using Laguerre operational matrices, to a matrix equation that can be solved by any suitable method. Comparison of the results with other methods such as Tau method [21] and Adomian decomposition method (ADM) [25] reveals that the Laguerre approach has more accuracy. In addition, to get the best approximating solution of the system, the truncation limit N must be chosen large enough. It is also to be mentioned that the method is efficient to determine the solution in closed form, as well.

Compliance with ethical standards
Conflict of interest All the authors declare that they have no conflict of interest.
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