The existence and uniqueness of solution for linear system of mixed Volterra-Fredholm integral equations in Banach space

Pakhshan M. Hasan; Nejmaddin A. Sulaiman; Fazlollah Soleymani; Ali Akgül; Pakhshan M. Hasan; Nejmaddin A. Sulaiman; Fazlollah Soleymani; Ali Akgül

doi:10.3934/math.2020014

AIMS Mathematics

2020, Volume 5, Issue 1: 226-235. doi: 10.3934/math.2020014

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Research article Special Issues

The existence and uniqueness of solution for linear system of mixed Volterra-Fredholm integral equations in Banach space

1.
Department of Mathematics, College of Education, Salahaddin University-Erbil, Iraq
2.
Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran
3.
Department of Mathematics, Art and Science Faculty, Siirt University, Siirt, Turkey

Received: 20 July 2019 Accepted: 15 October 2019 Published: 05 November 2019
MSC : 35A01, 32K05

In this paper, a linear system of mixed Volterra-Fredholm integral equations is considered. The problem of existence and uniqueness of its solution is investigated and proved in a complete metric space by using the Banach fixed-point theorem. Also, an iterative method of fixed point type is used to approximate the solution of the system. The algorithm is applied on several examples. To show the accuracy of the results and the efficiency of the method, the approximate solutions are compared with the exact solutions.

Keywords:

fixed point method,
contraction mapping,
Banach fixed-point (FP) theorem,
mixed Volterra-Fredholm integral equation

Citation: Pakhshan M. Hasan, Nejmaddin A. Sulaiman, Fazlollah Soleymani, Ali Akgül. The existence and uniqueness of solution for linear system of mixed Volterra-Fredholm integral equations in Banach space[J]. AIMS Mathematics, 2020, 5(1): 226-235. doi: 10.3934/math.2020014

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Abstract

A correction on

Numerical investigation and improvement of the aerodynamic performance of a modified elliptical-bladed Savonius-style wind turbine. By Sri Kurniati, Sudirman Syam and Arifin Sanusi. AIMS Energy, 2023, Volume 11, Issue 6: 1211–1230. Doi: 10.3934/energy.2023055

The authors would like to make the following corrections to the published paper [1].

On page 1213, we updated the contents of "one symbol statement: ρ" in section 2. The updated contents are as follows:

- ρ is the the density of air,

On page 1215, we updated the contents of "Eq 16" in section 2. The updated contents are as follows:

$\frac{{{\bf{∂}} {\bf{ \pmb{\mathsf{ ϕ}} }}}_{{\bf{ℓ}}}}{{\bf{∂}} \boldsymbol{t}}+{\bf{∇}} \left({{\varnothing }}_{{\bf{ℓ}}}\overrightarrow{\boldsymbol{V}}\right)-{\bf{∇}} \left({\bf{\Gamma }}_{{\bf{ℓ}}}\overrightarrow{\boldsymbol{V}}{{\varnothing }}_{{\bf{ℓ}}}\right) = {\boldsymbol{R}}_{{\bf{ℓ}}}$

(16)

On page 1216, we updated the contents of "Table 2" in section 2. The updated contents are as follows:

Table 2. The terms in the general transfer equation Eq 16.

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \frac{{\partial {\rm{ \mathsf{ ϕ} }}}_{\ell }}{\partial t}+\nabla \left({\varnothing }_{\ell }\overrightarrow{V}\right)-\nabla \left({\mathrm{\Gamma }}_{\ell }\overrightarrow{V}{\varnothing }_{\ell }\right)={R}_{\ell } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16)$
$l$	${\varnothing }_{\ell }$	${\mathrm{\Gamma }}_{\ell }$
1	U	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial x}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial x}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial x}\right)+gx$
2	V	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial y}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial y}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial y}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial y}\right)+gy$
3	W	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial z}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial z}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial z}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial z}\right)+gz$
4	1	0	0
5	K	$vt/$	$G-\varepsilon$
6	$\varepsilon$	$vt/$	${C}_{1}\frac{\varepsilon }{k}G-{c}_{2}\frac{\varepsilon }{k}\varepsilon$

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Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	A. M. Wazwaz, A reliable treatment for mix Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002), 405-414.
[2]	F. Mirzaee, S. F. Hoseini, Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations, Appl. Math. Comput., 273 (2016), 637-644.
[3]	F. Mirzaee, E. Hadadiyan, Numerical solution of Volterra-Fredholm integral equations via modification of hat functions, Appl. Math. Comput., 280 (2016), 110-123.
[4]	P. M. A. Hasan, N. A. Sulaiman, Existence and Uniqueness of Solution for Linear Mixed Volterra-Fredholm Integral Equations in Banach Space, Am. J. Comput. Appl. Math., 9 (2019), 1-5.
[5]	L. Mei, Y. Lin, Simplified reproducing kernel method and convergence order for linear Volterra integral equations with variable coefficients, J. Comput. Appl. Math., 346 (2019), 390-398. doi: 10.1016/j.cam.2018.07.027
[6]	S. Micula, On some iterative numerical methods for mixed Volterra-Fredholm integral equations, Symmetry, 11 (2019), 1200.
[7]	S. Deniz S, N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University - Science, 30 (2018), 91-99. doi: 10.1016/j.jksus.2016.09.001
[8]	K. Berrah, A. Aliouche, T. Oussaeif, Applications and theorem on common fixed point in complex valued b-metric space, AIMS Mathematics, 4 (2019), 1019-1033. doi: 10.3934/math.2019.3.1019
[9]	N. Bildik, S. Deniz, Solving the burgers' and regularized long wave equations using the new perturbation iteration technique, Numer. Meth. Part. D. E., 34 (2018), 1489-1501. doi: 10.1002/num.22214
[10]	J. Chen, M. He, Y. Huang, A fast multiscale Galerkin method for solving second order linear Fredholm integro-differential equation with Dirichlet boundary conditions, J. Comput. Appl. Math., 364 (2020), 112352.
[11]	R. Rabbani, R. Jamali, Solving nonlinear system of mixed Volterra- Fredholm integral equations by using variational iteration method, J. Math. Comput. Sci., 5 (2012), 280-287. doi: 10.22436/jmcs.05.04.05
[12]	M. Ghasemi, M. Fardi, R. K. Ghaziani, Solution of system of the mixed Volterra - Fredholm integral equations by an analytical method, Math. Comput. Model., 58 (2013), 1522-1530. doi: 10.1016/j.mcm.2013.06.006
[13]	A. Wazwaz, A First course in Integral Equations. Second Edition, Saint Xavier University, USA: World Scientific Publishing, 2015.
[14]	T. Abdeljawad, R. P. Agarwal, E. Karapınar, et al. Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space, Symmetry, 11 (2019), 686.
[15]	E. Hesameddini and M. Shahbazi, Solving system of Volterra-Fredholm integral equations with Bernstein polynomials and hybrid Bernstain Block pulse functions, J. Comput. Appl. Math., 315 (2017), 182-194. doi: 10.1016/j.cam.2016.11.004
[16]	A. Borhanifar, K. Sadri, Shifted Jacobi collocation method based on operational matrix for solving the systems of Fredholm and Volterra integral equations, Math. Comput. Appl., 20 (2015), 76-93.
[17]	R. V. Kakde, S. S. Biradar, S. S. Hiremath, Solution of Differential and Integral Equations Using Fixed Point Theory, International Journal of Advanced Research in Computer Engineering & Technology (IJARCET), 3 (2014), 1656-1659.
[18]	K. Maleknejad, P. Torabi, R. Mollapourasl, Fixed point method for solving nonlinear quadratic Volterra integral equations, Comput. Math. Appl., 62 (2011), 2555-2566. doi: 10.1016/j.camwa.2011.07.055
[19]	K. Maleknejad, P. Torabi, Application of fixed point method for solving nonlinear Volterra-Hammerstein integral, U. P. B. Sci. Bull., Series A: App. Math. Phy., 74 (2012), 45-56.
[20]	A. J. Jerri, Introduction to Integral Equation with Application. Marcel Dekker, New York and Basel, 1985.

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