Gegenbauer Polynomials Method is a new algorithm for solving linear and nonlinear integral equations using Galerkin weighted residual numerical method with Gegenbauer polynomials. Gegenbauer polynomials [1] is given as
3.6.1 Methodology
Definition 3.3. Consider the integral equation of the 1st kind is given as
where u(t) is the unknown function, to be determined, k(x,t), the kernel, is a continuous or discontinuous and square integrable function, f(x) being the known function.
Now we use the technique of Galerkin method, [Lewis, 2], to find an approximate solution 
of Eq. (3.79). For this, we assume that
where 
are Gegenbauer polynomials of degree defined in equation (3.78) and 
are unknown parameters, to be determined.
substituting (3.80) into (3.79)
we get,
Then the Galerkin equations are obtained by multiplying both sides of (3.81) by 
and then integrating with respect to x from a to b, we have
Since in each equation, there are two integrals. The inner integrand of the left side is a function of x, and t , and is integrated with respect to t from a to x . As a result the outer integrand becomes a function of only and integration with respect to x from a to b yields a constant. Thus for each j=0,1,2…,n we have a linear equation with n+1 unknowns ,k=0,1,2,…,n.
Finally (3.82) represents the system of n+1 linear equations in n+1 unknowns, are given by
where
Now the unknown parameters are determined by solving the system of equations (3.83) and substituting these values of parameters in (3.80), we get the approximate solution of the integral equation (3.79).
Definition 3.4. Consider the integral equation of the 2nd kind is
where u(t) is the unknown function, to be determined, k(x,t), the kernel, is a continuous or discontinuous and square integrable function,f(x) and u(x) being the known function and λ is a constant. Proceeding as before
where
Now the unknown parameters are determined by solving the system of equations (3.85) and substituting these values of parameters in (3.80), we get the approximate solution of the integral equation (3.84).
3.6.2 Applications of Gegenbauer Polynomials Method (GPM)
Example 3.23 Consider the Fredholm Integral equation [2]
The exact solution of Eq. (3.86) is
Consider 3rd order Gegenbauer Polynomials, i.e. for we have Eq. (3.86) is
Substituting Eq. (3.88) into Eq. (3.86)
Now multiply Eq. (3.89) by 
and integrating both side from -1 to 1
Now for β=1 the matrix form of Eq. (3.90) is
After solving we get
Consequently we have the approximate solution is
Consequently, we draw a table and graph.
TABLE 3.10 Comparison of the Exact solution and Approximate Solutions of Eq. (3.86) for n=3 using Gegenbauer Polynomials Method
Fig 3.10 Comparison of Exact and Approximate Solution of Eq. (3.86 ) for n=3.
Consider 4th order Gegenbauer Polynomials, i.e. for we have Eq. (3.86) is
Substituting Eq. (3.92) into Eq. (3.86)
Now multiply Eq. (16) by 
and integrating both side from -1 to 1, we have
Now for β=1 equation (3.94) can be written in the matrix form as,
After solving we get
Consequently we have approximate solution as
TABLE 3.11 Comparison of the Exact solution and Approximate Solutions of Eq. (3.86) for n=4 using Gegenbauer Polynomials Method
Fig 3.11 Comparison of Exact and Approximate Solutions of Eq. (3.86 ) for n=4.
Consider 5th order Gegenbauer Polynomials, i.e. for n=5 we have Eq. (3.86) is
Substituting Eq. (3.96) into Eq. (3.86)
Now multiply Eq. (3.97) by 
j=0,1,2,3,4,5 and integrating both side from -1 to 1, we have
Now for β=1 equation (3.98) can be written in the matrix form as,
After solving we get
After solving we get
TABLE 3.12 Comparison of the Exact solution and Approximate Solutions of Eq. (3.86) for n=5 using Gegenbauer Polynomials Method
Fig 3.12 Comparison of Exact and Approximate Solutions of Eq. (3.86 ) for n=5.
Example 3.24 Consider the Volterra Integral equation [2]
The exact solution of Eq. (3.100) is
According to the proposed technique, consider the trail solution
Consider 2nd order Gegenbauer Polynomials, i.e. for n=2 we have Eq. (3.102) is
Substituting Eq. (3.103) into Eq. (3.100)
Now multiply Eq. (3.104) by 
j=0,1,2 and integrating both side from -1 to 1, we have
Now for β=1 the matrix form of Eq. (3.105) is
After solving we get
Consequently we have the approximate solution is
Fig 3.13 Comparison of Exact and Approximate Solutions of Eq. (3.100 ) for n=2.
Table 3.13 Comparison of the Exact solution and Approximate Solutions of Eq. (3.100) for n=2 using Gegenbauer Polynomials Method.
Consider 4th order Gegenbauer Polynomials, i.e. for n=4 we have Eq. (3.102) is
Substituting Eq. (3.107) into Eq. (3.100)
Now multiply Eq. (3.108) by j=0,1,2,3,4 and integrating both side from -1 to 1, we have
Now for equation (3.109) can be written in the matrix form as,
After solving we get
Consequently we have the exact solution is
Fig 3.14 Comparison of Exact and Approximate Solutions of Eq. (3.100 ) for n=4.
Table 3.14 Comparison of the Exact solution and Approximate Solutions of Eq. (3.100) for n=4, using Gegenbauer Polynomials Method
Example 3.25 Consider the Abel’s Integral equation [2]
The exact solution of Eq. (3.111) is
The transformation
According to the proposed technique, consider the trail solution
Consider 1st order Gegenbauer Polynomials, i.e. for n=1, we have Eq. (3.114) is
Substituting Eq. (3.115) into Eq. (3.111)
Now multiply Eq. (3.116) by j= 0,1 and integrating both side from 0 to 1, we have
Now for β=1 the matrix form of Eq. (3.117) is
After solving we get
Consequently we have solution is
v(x) = x+3 (3.118)
This gives exact solution by
Obtained by using (3.113)
Fig 3.15 Comparison of Exact and Approximate Solutions of Eq. (3.111 ) for n=1.
TABLE 3.15 Comparison of the Exact solution and Approximate Solutions of Eq. (3.111) for n=1 using Gegenbauer Polynomials Method.
References
[1]. H.Exton, New generating function for Gegenbauer polynomials, J. Comput. App. Math, 67(1)( 1996), 191-193.
[2]. A.M. Wazwaz, Linear and Nonlinear Integral Equations Method and Applications, Springer Heidelberg Dordrecht London, New York, 2011.
[3]. A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).
[4]. Li Zhu, Qibin Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun Nonlinear Sci Numer Simulat 17 (2012), 2333–2341.