According to the HPM, we can use the embedding parameter as a “small parameter”, and assume that the solution of Equation (3) can be written as a power series in :
when , the approximate solution of Equation (2) is obtained with
The series Equation (6) is convergent for most cases; however, the rate of convergence depends upon the nonlinear operator L  .
3. The HPM Applied to Nonlinear Mixed V-FIE
To illustrate the HPM, for nonlinear mixed V-FIE let us consider the Equation (1)
By the HPM, we can expand into the form
and the approximate solution is
and in sum, according to  , He’s HPM considers the nonlinear term as
where are the so-called He’s polynomials  , which can be calculated by using the formula
Substituting (8) and (10) into (7) and equating the terms with identical powers of , we have
The components can be computing by using the recursive relations (12).
4. The Combined LHPM Applied to Nonlinear Mixed V-FIE 
We assume that the kernel of Equation (7) takes the form
Applying the Laplace transform to both sides of Equation (7), we represent the linear term from Equation (8) and the nonlinear term will be represented by the He’s polynomials from Equation (10) and equating the terms with identical powers of , we have:
Applying the inverse Laplace transform to the first part of Equation (13) gives , that will define . Utilizing will enable us to evaluate . The determination of and leads to the determination of that will allows us to determine , and so on. This in turn will lead to the complete determination of the components of , upon utilizing the second part of Equation (13). The series solution follows immediately after using Equation (9). The obtained series solution may converge to an exact solution if such a solution exists.
5. Numerical Examples
Example 5.1  :
Consider the linear mixed V-FIE with a generalized Cauchy kernel
we obtain Table 1.
Example 5.2  :
Consider the nonlinear mixed V-FIE with a generalized Cauchy kernel
we obtain Table 2.
The results for this examples using the LHPM obtained in Table 1 and Table 2 are best from the results in   where the solution was obtained using TMM.
Table 1. Results obtained for example 1 and error.
Table 2. Results obtained for example 2 and error.
In this article, we proposed LHPM and used it for solving nonlinear mixed V- FIE with a generalized singular kernel. As examples show, the displayed technique diminishes the computational difficulties of other methods. An interesting feature of this method is that the error is too small and all the calculations can be done straightforward. It can be concluded that LHPM is a very simple, powerful and effective method.
The authors would like to thank the king Abdulaziz city for science and technology.
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