Solvability and approximation of nonlinear functional mixed Volterra-Fredholm equation in Banach space. (English) Zbl 1515.65332

Summary: This study probes into the existence of a unique solution and the numerical approximation of a nonlinear functional Volterra-Fredholm integral equations of the mixed type and second kind. Based on the Lipschitz constants of the functional and kernel, a Bielecki’s norm is defined and used to modify a distance inequality on a constructed self-map. The map is shown to be contractive, thereby establishing solvability. The problem is then approximated by collocating at discrete points and use of a composite multidimensional numerical quadrature approximation. A new Grönwall-type inequality is proposed, and used, to prove the second order of convergence of the numerical scheme. Numerical experiments are provided to verify the theoretical results.


65R20 Numerical methods for integral equations
45G10 Other nonlinear integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
65D32 Numerical quadrature and cubature formulas


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[1] M. A. Abdou, “On a symptotic methods for Fredholm-Volterra integral equation of the second kind in contact problems”, J. Comput. Appl. Math. 154:2 (2003), 431-446. · Zbl 1019.45003 · doi:10.1016/S0377-0427(02)00862-2
[2] E. Alvarez and C. Lizama, “Application of measure of noncompactness to Volterra equations of convolution type”, J. Integral Equations Appl. 28:4 (2016), 441-458. · Zbl 1355.45005 · doi:10.1216/JIE-2016-28-4-441
[3] J. Banaś and B. Rzepka, “On existence and asymptotic stability of solutions of a nonlinear integral equation”, J. Math. Anal. Appl. 284:1 (2003), 165-173. · Zbl 1029.45003 · doi:10.1016/S0022-247X(03)00300-7
[4] S. Bazm, P. Lima, and S. Nemati, “Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type”, J. Comput. Appl. Math. 398 (2021), art. id. 113628. · Zbl 1472.65162 · doi:10.1016/j.cam.2021.113628
[5] A. Bielecki, “Une remarque sur la méthode de Banach-Cacciopoli-Tikhonov dans la théorie des équations différentielles ordinaires”, Bull. Acad. Polon. Sci. Cl. III. 4 (1956), 261-264. · Zbl 0070.08103
[6] Ü. Çakan, “On monotonic solutions of some nonlinear fractional integral equations”, Nonlinear Funct. Anal. Appl. 22:2 (2017), 259-273. · Zbl 1381.45021
[7] Ü. Çakan and İ. Özdemir, “An application of Krasnoselskii fixed point theorem to some nonlinear functional integral equations”, Nevşehir Bilim ve Teknol. Derg. 3:2 (2015), 66-73. · doi:10.17100/nevbiltek.210931
[8] A. Deep, A. Kumar, S. Abbas, and M. Rabbani, “Solvability and numerical method for non-linear Volterra integral equations by using Petryshyn’s fixed point theorem”, Int. J. Nonlinear Anal. Appl. 13:1 (2022), 1-28. · doi:10.22075/ijnaa.2021.22858.2422
[9] Deepmala and H. K. Pathak, “Study on existence of solutions for some nonlinear functional-integral equations with applications”, Math. Commun. 18:1 (2013), 97-107. · Zbl 1292.47063
[10] B. D. Dudson, J. Madsen, J. Omotani, P. Hill, L. Easy, and M. Løiten, “Verification of BOUT++ by the method of manufactured solutions”, Phys. Plasmas 23:6 (2016), art. id. 062303. · doi:10.1063/1.4953429
[11] P. M. Fitzpatrick, “Book review: nonlinear functional analysis”, Bull. Amer. Math. Soc. (N.S.) 20:2 (1989), 277-280.
[12] B. Grier, R. Figliola, E. Alyanak, and J. Camberos, “Discontinuous solutions using the method of manufactured solutions on finite volume solvers”, AIAA J. 53:8 (2015), 2369-2378. · doi:10.2514/1.J053725
[13] S. Hu, M. Khavanin, and W. Zhuang, “Integral equations arising in the kinetic theory of gases”, Appl. Anal. 34:3-4 (1989), 261-266. · Zbl 0697.45004 · doi:10.1080/00036818908839899
[14] A. J. Jerri, Introduction to integral equations with applications, 2nd ed., John Wiley & Sons, New York, 1999. · Zbl 0938.45001
[15] M. Kazemi, “Triangular functions for numerical solution of the nonlinear Volterra integral equations”, J. Appl. Math. Comput. 68:3 (2022), 1979-2002. · Zbl 07532913 · doi:10.1007/s12190-021-01603-z
[16] M. Kazemi and R. Ezzati, “Existence of solutions for some nonlinear Volterra integral equations via Petryshyn’s fixed point theorem”, Int. J. Nonlinear Anal. Appl. 9:1 (2018), 1-12. · Zbl 1412.47062
[17] M. Kazemi and A. R. Yaghoobnia, “Application of fixed point theorem to solvability of functional stochastic integral equations”, Appl. Math. Comput. 417 (2022), art. id. 126759. · Zbl 1510.60060 · doi:10.1016/j.amc.2021.126759
[18] M. Kazemi, H. M. Golshan, R. Ezzati, and M. Sadatrasoul, “New approach to solve two-dimensional Fredholm integral equations”, J. Comput. Appl. Math. 354 (2019), 66-79. · Zbl 1503.65319 · doi:10.1016/j.cam.2018.12.029
[19] T. D. Le, C. Moyne, M. A. Murad, and S. A. Lima, “A two-scale non-local model of swelling porous media incorporating ion size correlation effects”, J. Mech. Phys. Solids 61:12 (2013), 2493-2521. · Zbl 1294.76237 · doi:10.1016/j.jmps.2013.07.012
[20] N. Lungu and I. A. Rus, “On a functional Volterra-Fredholm integral equation, via Picard operators”, J. Math. Inequal. 3:4 (2009), 519-527. · Zbl 1195.45035 · doi:10.7153/jmi-03-51
[21] K. Maleknejad and M. Hadizadeh, “A new computational method for Volterra-Fredholm integral equations”, Comput. Math. Appl. 37:9 (1999), 1-8. · Zbl 0940.65151 · doi:10.1016/S0898-1221(99)00107-8
[22] K. Maleknejad, R. Mollapourasl, and K. Nouri, “Study on existence of solutions for some nonlinear functional-integral equations”, Nonlinear Anal. 69:8 (2008), 2582-2588. · Zbl 1156.45006 · doi:10.1016/j.na.2007.08.040
[23] S. Micula, “On some iterative numerical methods for mixed Volterra-Fredholm integral equations”, Symmetry 11:10 (2019), art. id. 1200. · doi:10.3390/sym11101200
[24] F. Mirzaee and N. Samadyar, “Extension of Darbo fixed-point theorem to illustrate existence of the solutions of some nonlinear functional stochastic integral equations”, Int. J. Nonlinear Anal. Appl. 11:1 (2020), 413-421. · Zbl 1516.47124 · doi:10.22075/ijnaa.2018.14031.1743
[25] C. Nwaigwe, Coupling methods for 2D/1D shallow water flow models for flood simulations, Ph.D. thesis, University of Warwick, November 2016, available at http://wrap.warwick.ac.uk/88277/.
[26] C. Nwaigwe, “Analysis and application of a convergent difference scheme to nonlinear transport in a Brinkman flow”, Int. J. Numer. Methods Heat Fluid Flow 30:30 (2020), 4453-4473. · doi:10.1108/HFF-10-2019-0758
[27] W. L. Oberkampf and T. G. Trucano, “Verification and validation benchmarks”, Nucl. Eng. Des. 238:3 (2008), 716-743. · doi:10.1016/j.nucengdes.2007.02.032
[28] D. O’Regan, “Existence results for nonlinear integral equations”, J. Math. Anal. Appl. 192:3 (1995), 705-726. · Zbl 0851.45003 · doi:10.1006/jmaa.1995.1199
[29] I. Özdemir and U. Çakan, “On the solutions of a class of nonlinear functional integral equations in space \[C[0,a]\]”, J. Math. Appl. 38 (2015), 105-114. · Zbl 1382.45011
[30] I. Özdemir, U. Çakan, and B. İlhan, “On the existence of the solutions for some nonlinear Volterra integral equations”, Abstr. Appl. Anal. (2013), art. id. 698234. · Zbl 1470.45008 · doi:10.1155/2013/698234
[31] J. Prüss, Evolutionary integral equations and applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993. · Zbl 0784.45006 · doi:10.1007/978-3-0348-8570-6
[32] M. Rabbani, “An iterative algorithm to find a closed form of solution for Hammerstein nonlinear integral equation constructed by the concept of cosm-rs”, Math. Sci. 13:3 (2019), 299-305. · Zbl 1447.45007 · doi:10.1007/s40096-019-00299-4
[33] P. J. Roache, “The method of manufactured solutions for code verification”, pp. 295-318 in Computer simulation validation: fundamental concepts, methodological frameworks, and philosophical perspectives, Springer, Cham, 2019. · Zbl 1410.68010 · doi:10.1007/978-3-319-70766-2_12
[34] A. C. Rocha, M. A. Murad, C. Moyne, S. P. Oliveira, and T. D. Le, “A new methodology for computing ionic profiles and disjoining pressure in swelling porous media”, Comput. Geosci. 20:5 (2016), 975-996. · Zbl 1391.76517 · doi:10.1007/s10596-016-9572-5
[35] L. Wang, W.-j. Zhou, and C.-q. Ji, “Verification of a chemical nonequilibrium flows solver using the method of manufactured solutions”, Procedia Eng. 99 (2015), 713-722. · doi:10.1016/j.proeng.2014.12.593
[36] J. Wang, W. Martin, and B. Collins, “The application of Method of Manufactured Solutions to method of characteristics in planar geometry”, Ann. Nucl. Energy 121 (2018), 295-304. · doi:10.1016/j.anucene.2018.07.041
[37] A.-M. Wazwaz, “A reliable treatment for mixed Volterra-Fredholm integral equations”, Appl. Math. Comput. 127:2-3 (2002), 405-414. · Zbl 1023.65142 · doi:10.1016/S0096-3003(01)00020-0
[38] A.-M. Wazwaz, A first course in integral equations, 2nd ed., World Scientific Publishing Company, 2015. · Zbl 1332.45001
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