Using of Bernstein spectral Galerkin method for solving of weakly singular Volterra–Fredholm integral equations

In this work, we applied a new method for solving the linear weakly singular mixed Volterra–Fredholm integral equations. We now begin the theoretical study with acquirement of the variational form; in addition, we are using Bernstein spectral Galerkin method to be approximate to my problems. We estimate the error of the method by proved some theorems. Moreover, in the final section, we solved some numerical examples.


Introduction
One of the best subjects in the numerical analysis is a finite-element method (FEM). We used (FEM) to solve problems in mathematical physics, integral equations, and engineering, such as electromagnetic potential, fluid flow, radiation heats transfer, laminar boundary-layer theory and mass transport, Abel integral equations, and problem of mechanics or physics [3-5, 7, 18, 20]. For approximating of singular or weakly singular integral equations, there are several numerical method's existences. For example, Baratella and Orsi [8] were introduced weakly singular for Volterra, and discussed on operational matrix method with block-pulse functions by Babolian and Salimi [6]. Furthermore, some author works to be approximate to Abel integral equations, for example, Garza in [12] and Hall in [13] used the wavelet method, Legendre wavelets approximation by Yousefi in [19], Gauss-Jacobi quadrature rule by Fettis in [10], and Piessens and Verbaeten in [16,17] with Chebyshev polynomials of the first kind.
In this paper, we use (FEM) and Bernstein polynomials to acquire an approximate solution for linear weakly singular mixed Volterra-Fredholm integral equation as follows: Therefore, we can be plotted this ten first Bernstein basis polynomials on the unit square as follows ( Fig. 1):

Implementation of the Bernstein-Galerkin method
In this section with using of Galerkin method, we find an approximate solution for Eq. (1). For this purpose, we obtain weak and variational form. then, B is a bilinear form: Bðk 1 u þ k 2 w; vÞ ¼ k 1 Bðu; vÞ þ k 2 Bðw; vÞ: We consider that V h ¼ spanf/ 1 ; / 2 ; . . .; / n g is a subspace of V, and f/ i g n i¼1 are a set of Bernstein polynomial functions of degree at most m in each subinterval. Hence, by substituting (4) in variational formulation, we have Now, for i; j ¼ 1; 2; . . .; n; we define and From system (8), we have . . .; a n T ; G ¼ ½G 1 ; G 2 ; . . .; G n T ; . . .; n: By solving of the system (9), we can obtain approximate solution of Eq. (1).

Error analysis
In this section, using the theorem, we get an upper bound for the error of our method, and we proved that the order of convergence is a Oðh f Þ. For this purpose, suppose that V and B are a Hilbert space and symmetric, respectively.
Definition 1 If B is a V-elliptic bilinear form, then an inner product energy is a ð:; :Þ : V Â V ! R and the energy norm as  (3) is a V-ellipticity and Eq.
(1) has a unique solution, and order of convergence is a Oðh f Þ.
with using of the Cauchy-Schwarz inequality and L 2 -norm, we have then Bðv; vÞ ! ðgÞjjvjj 2 L 2 ðXÞ ; ð12Þ thus B is a V-ellipticity; therefore, using of Lax-Milgram theorem and V-ellipticity of B, Eq. (1) has a unique solution. Suppose u h is an approximate solution, so we have and If e ¼ u À u h that u are an exact solution of Eq. (1), then By Schwartz's inequality, and relation between energy norm and inner product, we have jBðv; wÞj jjvjj E jjwjj E ; 8v; w 2 V: ð16Þ Using (15), we have Therefore, e is an orthogonal for any v h . Using and Cea's Lemma [9], for each particularṽ h in V h , we have ifṽ h is equal toũ h , then and the exact solution is a uðxÞ ¼ x 2 ð1 À xÞ.
With using Bernstein basis polynomials of degree 2, and M ¼ 5, the results of obtained are presented in Table 1 and In addition, the exact solution is uðxÞ ¼ x.
With using of Bernstein basis polynomials of degree 2, and M ¼ 5, the results of obtained are presented in Table 2 and Fig. 3.  ffiffiffiffiffiffiffiffiffiffi ffi ; and the exact solution is uðxÞ ¼ expðxÞ: With using of Bernstein basis polynomials of degree 2, and M ¼ 5, the results of obtained are presented in Table 3 and Fig. 4.

Conclusions
In this paper, we used of Galerkin method and Bernstein polynomials to solving one of the most important linear weakly singular Volterra-Fredholm integral equation, with     Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creative commons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.