An effective computational approach based on Hermite wavelet Galerkin for solving parabolic Volterra partial integro differential equations and its convergence analysis
Abstract
In this research article Hermite wavelet based Galerkin method is developed for the numerical solution of Volterra integro-differential equations in onedimension with initial and boundary conditions. These equations include the partial differential of an unknown function and the integral term containing the unknown function which is the memory of the problem. Wavelet analysis is a recently developed mathematical tool in applied mathematics. For this purpose, Hermit wavelet Galerkin method has proven a very powerful numerical technique for the stable and accurate solution of giving boundary value problem. The theorem of convergence analysis and compare some numerical examples with the use of the proposed method and the exact solutions shows the efficiency and high accuracy of the proposed method. Several figures are plotted to establish the error analysis of the approach presented.
Keyword : Volterra partial integro-differential equation, Hermite wavelet, Galerkin method
How to Cite
Rostami, Y. (2023). An effective computational approach based on Hermite wavelet Galerkin for solving parabolic Volterra partial integro differential equations and its convergence analysis. Mathematical Modelling and Analysis, 28(1), 163–179. https://doi.org/10.3846/mma.2023.15690
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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R.K. Sinha and B. Deka. A prior eroor estimates in finite element method for nonselfadjoint elliptic and parabolic interface problems. Calcolo, 43:253–278, 2006. https://doi.org/10.1007/s10092-006-0122-8
Z. Sun and X. Wu. A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math., 56(2):193–209, 2006. https://doi.org/10.1016/j.apnum.2005.03.003
Y. Yan and G. Fairweather. Orthogonal spline collocation methods for some partial integrodifferential equations. SIAM J. Numer. Anal., 29(3):755–768, 1992. https://doi.org/10.1137/0729047
E.G. Yanik and G. Fairweather. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal., 12(8):785–809, 1988. https://doi.org/10.1016/0362-546X(88)90039-9
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K.S. Zadeh. An integro-partial differential equation for modeling biofluids flow in fractured biomaterials. Theoretical Biology, 273(1):72–79, 2011. https://doi.org/10.1016/j.jtbi.2010.12.039
E. Zeidler. Nonlinear Functional Analysis and its Application-Linear Monotone Operators.Springer,NewYork,1990. https://doi.org/10.1007/978-1-4612-0981-2
A. Ali, M.A. Iqbal and S.T. Mohyud-Din. Hermites wavelets method for boundary value problems. International Journal of Modern Applied Physics, 3(1):38– 47, 2013.
K. Atkinson and A. Bogomolny. The discrete Galerkin method for integral equations. Mathematics of computation, 48(178):595–616, 1987. https://doi.org/10.1090/S0025-5718-1987-0878693-6
Z. Avazzadeh, Z. Beygi Rizi, F.M. Maalek Ghaini and G.B. Loghmani. A numerical solution of nonlinear parabolic-type Volterra partial integro-differential equations using radial basis functions. Engineering Analysis with Boundary Elements, 36(5):881–893, 2012. https://doi.org/10.1016/j.enganabound.2011.09.013
F. Fakhar-Izadi and M. Dehghan. The spectral methods for parabolic Volterra integro-differential equations. J Comput Appl Math, 235(14):4032–4046, 2011. https://doi.org/10.1016/j.cam.2011.02.030
J. Guo, D. Xu and W. Qiu. A finite difference scheme for the nonlinear timefractional partial integro-differential equation. Mathematical Methods in Applied Sciences, 43(6):3392–3412, 2020. https://doi.org/10.1002/mma.6128
P. Hepperger. Hedging electricity swaptions using partial integro-differential equations. Stochastic Processes And Their Applications., 122(2):600–622, 2012. https://doi.org/10.1016/j.spa.2011.09.005
J.-P. Kauthen. The method of lines for parabolic partial integrodifferential equations. J Integr Equat Appl, 4(1):69–81, 1992. https://doi.org/10.1216/jiea/1181075666
S. Kumbinarasaiah and R.A. Mundewadi. The new operational matrix of integration for the numerical solution of integro-differential equations via Hermite wavelet. SeMA Journal, (78):367–384, 2020. https://doi.org/10.1007/s40324-020-00237-8
J.L. Lions and E. Magenes. Nonohomogeneous Boundary Value Problems and Applications. Springer, Berlin, 1972. https://doi.org/10.1007/978-3-642-65217-2
K. Maleknejad and A. Ebrahimzadeh. The use of rationalized Haar wavelet collocation method for solving optimal control of Volterra integral equations. Journal of Vibration and Control, 21(10):1958–1967, 2015. https://doi.org/10.1177/1077546313504977
R.A. Mundewadi and B.A. Mundewadi. Hermite wavelet collocation method for the numerical solution of integral and integro-differential equations. International Journal of Mathematics Trends and Technology, 53(3):215–231, 2018. https://doi.org/10.14445/22315373/IJMTT-V53P527
O. Oruc¸. An efficient wavelet collocation method for nonlinear two-space di-¨ mensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation. Engineering with Computers, 36:839–856, 2019. https://doi.org/10.1007/s00366-019-00734-z
O. Oruc¸, F. Bulut and A. Esen. Numerical solutions of regularized long wave¨ equation by Haar wavelet method. Mediterranean Journal of Mathematics, 13:3235–3253, 2016. https://doi.org/10.1007/s00009-016-0682-z
O. Oruc¸, F. Bulut and A. Esen. Chebyshev wavelet method for numerical so-¨ lutions of coupled Burgers’ equation. Hacettepe Journal of Mathematics and Statistics, 48(1):1–16, 2019. https://doi.org/10.15672/HJMS.2018.642
M. Dehghan P. Assari. A local Galerkin integral equation method for solving integro-differential equations arising in oscillating magnetic fields. Mediterranean Journal of Mathematics, 90, 2018.
B.G. Pachpatte. On a nonlinear diffusion system arising in reactor dynamics. Math Analysis Applic, 94(2):501–508, 1983. https://doi.org/10.1016/0022-247X(83)90078-1
J. Petrolito. Approximate solutions of differential equations using Galerkin’s method and weighted residuals. International Journal of Mechanical Engineering Education, 28(1):14–26, 2000. https://doi.org/10.7227/IJMEE.28.1.2
E.W. Sachs and A.K. Strauss. Efficient solution of a partial integro-differential equation in finance. Applied Numerical Mathematics, 58(11):1687–1703, 2008. https://doi.org/10.1016/j.apnum.2007.11.002
P.K. Sahu and S. Saha Ray. Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Applied Mathematics and Computation, 256:715–723, 2015. https://doi.org/10.1016/j.amc.2015.01.063
S.C. Shiralashetti and S. Kumbinarasaiah. Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems. Alexandria Engineering Journal, 57(4):2591–2600, 2018. https://doi.org/10.1016/j.aej.2017.07.014
G. Singh and I. Singh. Solving some differential equations arising in electric engineering using Hermite polynomials. Journal of Scientific Research, 12(4):517– 523, 2020. https://doi.org/10.3329/jsr.v12i4.45686
R.K. Sinha and B. Deka. A prior eroor estimates in finite element method for nonselfadjoint elliptic and parabolic interface problems. Calcolo, 43:253–278, 2006. https://doi.org/10.1007/s10092-006-0122-8
Z. Sun and X. Wu. A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math., 56(2):193–209, 2006. https://doi.org/10.1016/j.apnum.2005.03.003
Y. Yan and G. Fairweather. Orthogonal spline collocation methods for some partial integrodifferential equations. SIAM J. Numer. Anal., 29(3):755–768, 1992. https://doi.org/10.1137/0729047
E.G. Yanik and G. Fairweather. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal., 12(8):785–809, 1988. https://doi.org/10.1016/0362-546X(88)90039-9
V.F. Kovalev Y.N. Grigoriev, N.H. Ibragimov and S.V. Meleshko. Symmetries of integro-differential equations: With applications in mechanics and plasma physics. Springer., 2010.
K.S. Zadeh. An integro-partial differential equation for modeling biofluids flow in fractured biomaterials. Theoretical Biology, 273(1):72–79, 2011. https://doi.org/10.1016/j.jtbi.2010.12.039
E. Zeidler. Nonlinear Functional Analysis and its Application-Linear Monotone Operators.Springer,NewYork,1990. https://doi.org/10.1007/978-1-4612-0981-2