International Journal of Mathematics Trends and Technology
Volume 53 | Number 3 | Year 2018 | Article Id. IJMTT-V53P527 | DOI : https://doi.org/10.14445/22315373/IJMTT-V53P527
Hermite wavelet collocation method for the numerical solution of Volterra, Fredholm, mixed Volterra-Fredholm integral equations, integro-differential equations and Abel’s integral equations. The method is based upon Hermite polynomials and Hermite wavelet approximations. The properties of Hermite wavelet is first presented and the resulting Hermite wavelet matrices are utilized to reduce the integral and integro-differential equations into system of algebraic equations to get the required Hermite coefficients are computed using Matlab. This technique is tested, some numerical examples and compared with the exact and existing method. Error analysis is worked out, which shows the efficiency of the proposed method.
[1] Chui C. K., Wavelets: A mathematical tool for signal analysis, in: SIAM Monographs on Mathematical Modeling and Computation, SIAM, Philadelphia, Pennsylvania, 1997.
[2] Beylkin G., Coifman R. and Rokhlin V., Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math., 44, 1991, 141–183.
[3] Lepik Ü. and Tamme E., Application of the Haar wavelets for solution of linear integral equations, Ant. Turk–Dynam. Sys. Appl. Proce., 2005, 395–407.
[4] Lepik Ü., Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul., 68, 2005, 127-143.
[5] Lepik Ü. and Tamme E., Application of the Haar wavelets for solution of linear integral equations, Ant. Turk–Dynam. Sys. Appl. Proce., 2005, 395–407.
[6] Lepik Ü., Haar wavelet method for nonlinear integro-differential equations, Appl. Math. Comp., 176, 2006, 324-333.
[7] Lepik Ü., Application of the Haar wavelet transform to solving integral and differential Equations, Proc. Estonian Acad Sci. Phys. Math., 56, 2007, 28-46.
[8] Lepik Ü. and Tamme E., Solution of nonlinear Fredholm integral equations via the Haar wavelet method, Proc. Estonian Acad. Sci. Phys. Math., 56, 2007, 17–27.
[9] Lepik Ü., Numerical solution of evolution equations by the Haar wavelet method, Appl. Math. Comput. 185, 2007, 695–704.
[10] Lepik Ü., Solving fractional integral equations by the Haar wavelet method, Appl. Math. Comp., 214, 2009, 468-478.
[11] Maleknejad K. and Mirzaee F., Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method, 80(11), 2003, 1397-1405.
[12] Maleknejad K. and Mirzaee F., Using rationalized haar wavelet for solving linear integral equations, App. Math. Comp., 160, 2005, 579 – 587.
[13] Maleknejad K., Kajani M. T. and Mahmoudi Y., Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelets, J. Kybernet., 32, 2003, 1530-1539.
[14] Maleknejad K. and Yousefi M., Numerical solution of the integral equation of the second kind by using wavelet bases of Hermite cubic splines, App. Math. Comp., 183, 2006, 134-141.
[15] Maleknejad K., Lotfi T. and Rostami Y., Numerical computational method in solving Fredholm integral equations of the second kind by using Coifman wavelet, App. Math. Comp. 186, 2007, 212-218.
[16] Babolian E. and Fattahzadeh F., Numerical computational method in solving integral equations by using chebyshev wavelet operational matrix of integration, Appl. Math. Comp., 188, 2007, 1016 -1022.
[17] Liang X. Z., Liu M. and Che X., Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets, J. Comp. App. Math., 136, 2001, 149-161.
[18] Yousefi S. and Banifatemi A., Numerical solution of Fredholm integral equations by using CAS wavelets, App. Math. Comp., 183, 2006, 458-463.
[19] Gao J. and Jiang Y., Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel, J. Comp. Appl. Math., 215, 2008, 242 – 259.
[20] Abdalrehman A. A., An Algorithm for nth Order Intgro-Differential Equations by Using Hermite Wavelets Functions, Bagh. Sci. Jour., 11(3) 2014.
[21] Ali A., Asad Iqbal M. and Mohyud-Din S. T., Hermite Wavelets Method for Boundary Value Problems, Inter. J. Mod. Appl. Phys., 3(1), 2013, 38-47.
[22] Saeed U. and Rehman M., Hermite Wavelet Method for Fractional Delay Differential Equations, J. Diff. Equ., 2014, http://dx.doi.org/10.1155/2014/359093.
[23] Blyth W. F., May R. L. and Widyaningsih P., Volterra integral equations solved in fredholm form using walsh functions, ANZIAM J., 45, 2004, C269-C282.
[24] Wazwaz A. M., Linear and nonlinear integral equations methods and applications, Springer, 2011.
[25] Sahu P. K. and Ray S. S., Legendre wavelets operational method for the numerical solution of nonlinear volterra integro-differential equations system, Appl. Math. Comp., 256, 2015, 715-723.
[26] Mirzaee F., A Computational Method for Solving Linear Volterra Integral Equations, Appl. Math. Sci., 6, 2012, 807 – 814.
[27] Danfu H. and Xufeng S., Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration, Appl. Math. Comp., 194, 2007, 460–466.
[28] Wazwaz A. M., A First course in integral equations, World Scientific, 2015.
[29] Jahanshahi S., Babolian E., Torres D. F. M., Solving Abel integral equations of first kind via fractional calculus, J. king saud Uni. Sci., 27,, 2015, 161-167.
[30] Sohrabi S., Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation, Ain Shams Eng. J., 2, 2011, 249–254.
[31] Ramane H. S., Shiralashetti S. C., Mundewadi R. A., Jummannaver R. B., Numerical solution of Fredholm Integral Equations Using Hosoya Polynomial of Path Graphs, American Journal of Numerical Analysis, 5(1), 2017, 11-15.
Ravikiran A. Mundewadi, Bhaskar A. Mundewadi, "Hermite Wavelet Collocation Method for the Numerical Solution of Integral and Integro - Differential Equations," International Journal of Mathematics Trends and Technology (IJMTT), vol. 53, no. 3, pp. 215-231, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V53P527
PDF 
Abstract 
Keywords 
References 
Citation