This study is focused on the numerical solutions of the nonlinear Volterra-Fredholm integral equations (NV-FIEs) of the second kind, which have several applications in physical mathematics and contact problems. Herein, we develop a new technique that combines the modified Adomian decomposition method and the quadrature (trapezoidal and Weddle) rules that used when the definite integral could be extremely difficult, for approximating the solutions of the NV-FIEs of second kind with a phase lag. Foremost, Picard's method and Banach's fixed point theorem are implemented to discuss the existence and uniqueness of the solution. Furthermore, numerical examples are presented to highlight the proposed method's effectiveness, wherein the results are displayed in group of tables and figures to illustrate the applicability of the theoretical results.
Citation: Gamal A. Mosa, Mohamed A. Abdou, Ahmed S. Rahby. Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag[J]. AIMS Mathematics, 2021, 6(8): 8525-8543. doi: 10.3934/math.2021495
This study is focused on the numerical solutions of the nonlinear Volterra-Fredholm integral equations (NV-FIEs) of the second kind, which have several applications in physical mathematics and contact problems. Herein, we develop a new technique that combines the modified Adomian decomposition method and the quadrature (trapezoidal and Weddle) rules that used when the definite integral could be extremely difficult, for approximating the solutions of the NV-FIEs of second kind with a phase lag. Foremost, Picard's method and Banach's fixed point theorem are implemented to discuss the existence and uniqueness of the solution. Furthermore, numerical examples are presented to highlight the proposed method's effectiveness, wherein the results are displayed in group of tables and figures to illustrate the applicability of the theoretical results.
| [1] |
S. Abbasbandy, E. Shivanian, A new analytical technique to solve Fredholm's integral equations, Numer. Algorithms, 56 (2011), 27–43. doi: 10.1007/s11075-010-9372-2
|
| [2] | M. A. Abdou, M. M. El-Kojok, Numerical method for the two-dimensional mixed nonlinear integral equation in time and position, Univers. J. Integr. Equations, 4 (2016), 42–53. |
| [3] |
M. A. Abdou, S. A. Raad, New numerical approach for the nonlinear quadratic integral equations, J. Comput. Theor. Nanosci., 13 (2016), 6435–6439. doi: 10.1166/jctn.2016.5582
|
| [4] |
A. Akbarzadeh, J. Fu, Z. Chen, Three-phase-lag heat conduction in a functionally graded hollow cylinder, Trans. Can. Soc. Mech. Eng., 38 (2014), 155–171. doi: 10.1139/tcsme-2014-0010
|
| [5] | H. Almasieh, J. Meleh, Hybrid functions method based on radial basis functions for solving nonlinear Fredholm integral equations, J. Math. Ext., 7 (2014), 29–38. |
| [6] |
P. Assari, H. Adibi, M. Dehghan, A meshless method based on the moving least squares (mls) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains, Numer. Algorithms, 67 (2014), 423–455. doi: 10.1007/s11075-013-9800-1
|
| [7] | K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge: Cambridge University Press, 1997. |
| [8] |
I. Aziz, New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math., 239 (2013), 333–345. doi: 10.1016/j.cam.2012.08.031
|
| [9] | C. Brezinski, M. Redivo-Zaglia, Extrapolation methods for the numerical solution of nonlinear Fredholm integral equations, J. Integr. Equations Appl., 31 (2019), 29–57. |
| [10] |
H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal., 27 (1990), 987–1000. doi: 10.1137/0727057
|
| [11] |
P. Cheng, J. Huang, Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods, Numer. Algorithms, 58 (2011), 545–554. doi: 10.1007/s11075-011-9469-2
|
| [12] |
S. Chiriţă, On the time differential dual-phase-lag thermoelastic model, Meccanica, 52 (2017), 349–361. doi: 10.1007/s11012-016-0414-2
|
| [13] | C. Constanda, M. E. Pérez, Integral Methods in Science and Engineering, Springer, 2010. |
| [14] | M. M. El-Borai, M. A. Abdou, M. M. El-Kojok, On a discussion of nonlinear integral equation of type Volterra-Fredholm, J. Korean Soc. Ind. Appl. Math., 10 (2006), 59–83. |
| [15] |
J. A. Ezquerro, M. A. Hernández-Verón, Nonlinear Fredholm integral equations and majorant functions, Numer. Algorithms, 82 (2019), 1303–1323. doi: 10.1007/s11075-019-00656-3
|
| [16] |
J. Gao, M. Condon, A. Iserles, Spectral computation of highly oscillatory integral equations in laser theory, J. Comput. Phys., 395 (2019), 351–381. doi: 10.1016/j.jcp.2019.06.045
|
| [17] |
F. Ghoreishi, M. Hadizadeh, Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations, Numer. Algorithms, 52 (2009), 541. doi: 10.1007/s11075-009-9297-9
|
| [18] |
A. Hadjadj, J. Dussauge, Shock wave boundary layer interaction, Shock Waves, 19 (2009), 449–452. doi: 10.1007/s00193-009-0238-2
|
| [19] | A. Jerri, Introduction to Integral Equations with Applications, John Wiley & Sons, 1999. |
| [20] |
R. Katani, Numerical solution of the Fredholm integral equations with a quadrature method, SeMA J., 76 (2019), 271–276. doi: 10.1007/s40324-018-0175-z
|
| [21] |
F. R. Lin, Preconditioned iterative methods for the numerical solution of Fredholm equations of the second kind, Calcolo, 40 (2003), 231–248. doi: 10.1007/s10092-003-0078-x
|
| [22] | K. Maleknejad, M. Hadizadeh, A new computational method for Volterra-Fredholm integral equations, Comput. Math. Appl., 37 (1999), 1–8. |
| [23] |
S. Noeiaghdam, M. A. F. Araghi, S. Abbasbandy, Optimal convergence control parameter in the homotopy analysis method to solve integral equations based on the stochastic arithmetic, Numer. Algorithms, 81 (2019), 237–267. doi: 10.1007/s11075-018-0546-7
|
| [24] |
S. Pishbin, Numerical solution and structural analysis of two-dimensional integral-algebraic equations, Numer. Algorithms, 73 (2016), 305–322. doi: 10.1007/s11075-016-0096-9
|
| [25] | A. B. Sawaoka, Shock Waves in Materials Science, Springer Science & Business Media, 2012. |
| [26] |
H. Song, Z. Yang, H. Brunner, Analysis of collocation methods for nonlinear Volterra integral equations of the third kind, Calcolo, 56 (2019), 7. doi: 10.1007/s10092-019-0304-9
|
| [27] | K. Takayama, Shock Waves: Proceedings of the 18th International Symposium on Shock Waves, Held at Sendai, Japan 21–26 July 1991, Springer Science & Business Media, 2012. |
| [28] | A. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002), 405–414. |
| [29] | A. Wazwaz, Linear and Nonlinear Integral Equations, Vol. 639, Berlin: Springer, 2011. |
| [30] | A. Wazwaz, S. M. El-Sayed, A new modification of the Adomian decomposition method for linear and nonlinear operators, Appl. Math. Comput., 122 (2001), 393–405. |
| [31] |
J. Xie, X. Gong, W. Shi, R. Li, W. Zhao, T. Wang, Applying the three-dimensional block-pulse functions to solve system of Volterra-Hammerstein integral equations, Numer. Methods Partial Differ. Equations, 36 (2020), 1648–1661. doi: 10.1002/num.22496
|
| [32] |
J. Xie, Q. Huang, F. Zhao, Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations in two-dimensional spaces based on block pulse functions, J. Comput. Appl. Math., 317 (2017), 565–572. doi: 10.1016/j.cam.2016.12.028
|
| [33] |
J. Xie, Z. Ren, Y. Li, X. Wang, T. Wang, Numerical scheme for solving system of fractional partial differential equations with Volterra-type integral term through two-dimensional block-pulse functions, Numer. Methods Partial Differ. Equations, 35 (2019), 1890–1903. doi: 10.1002/num.22383
|
| [34] | J. Xie, M. Yi, Numerical research of nonlinear system of fractional Volterra-Fredholm integral-differential equations via block-pulse functions and error analysis, J. Comput. Appl. Math., 34 (2019), 159–167. |
| [35] | S. M. Zemyan, The Classical Theory of Integral Equations: A Concise Treatment, Springer Science & Business Media, 2012. |
| [36] | F. Zhang, Shock Waves Science and Technology Library, Detonation Dynamics, Springer Science & Business Media, 2012. |
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Gamal A. Mosa, Mohamed A. Abdou, Ahmed S. Rahby. Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag[J]. AIMS Mathematics, 2021, 6(8): 8525-8543. doi: 10.3934/math.2021495
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