Galerkin Finite Element Methods Error Estimates Analysis for [1+2] Equations of Integro-Differential on Linear Triangular
DOI:
https://doi.org/10.21123/bsj.2024.8848Keywords:
Backward-Euler, Parabolic Integro-Differential, Quadrature procedures, Two-dimensional, Volterra integral termAbstract
In this research, the two-dimensional parabolic integral-differential equation was solved using one of the numerical methods, which is the finite element method (Galerkin) on triangular elements. This method was chosen to make extensive use of finite elements because it has many high-quality numerical properties. The main benefit of finite elements is their ability to solve a wide range of problems in different computational fields in different forms, especially complex ones that cannot be solved by other numerical methods. Given the semi-discrete error estimates for the normal space , the polynomial linear boundary element space defined in triangles was used to describe space and the inverse Euler method was used to describe time. The discriminant rules used to differentiate the Volterra integral term are also chosen to be compatible with time phase diagrams. In addition, the numerical solutions of the two-dimensional differential integral equation of the equivalent type are compared with the exact solutions, and finally the final results of the solutions are displayed graphically using MATLAB. Finite element Galerkin error analysis was taken into account when using a mesh of triangular elements on the differential equation in two-dimensional space.
Received 04/04/2023
Revised 05/03/2024
Accepted 07/03/2024
Published Online First 20/09/2024
References
Courant R. Variational methods for the solution of problems equilibrium and vibrations. Bull Amer Math Soc. 1943; 49: 1–23. http://dx.doi.org/10.1090/S0002-9904-1943-07818-4
Zienkiewicz O, Taylor RL, Zhu JZ. The Finite Element Method: Its Basis and Fundamentals. 7th edition. UK: Butterworth-Heinemann; 2013. 704 p. https://doi.org/10.1016/c2009-0-24909-9
Wang P, Huo J, Wang X-M, Wang B-H. Diffusion and memory effect in a stochastic process and the correspondence to an information propagation in a social system. Physica A Stat Mech Appl. 2022; 607: 128206. https://doi.org/10.1016/j.physa.2022.128206
Hosseini-Farid M, Ramzanpour M, McLean J, Ziejewski M, Karami G. A poro-hyper-viscoelastic rate-dependent constitutive modeling for the analysis of brain tissues. J Mech Behav Biomed Mater. 2020; 102: 103475. https://doi.org/10.1016/j.jmbbm.2019.103475
Han H, Zhang C. Asymptotical Stability of Neutral Reaction-Diffusion Equations with PCAS and Their Finite Element Methods. Acta Math Sci. 2023; 43: 1865–1880. https://doi.org/10.1007/s10473-023-0424-9
Barbeiro S, Ferreira JA. Integro-differential models for percutaneous drug absorption. Int J Comput Math. 2007; 84(4): 451–467. https://doi.org/10.1080/00207160701210091
Hussain KH, Hamoud AA, Mohammed NM. Some New Uniqueness Results for Fractional Integro-Differential Equations. Nonlinear Funct. Anal Appl. 2019; 24(4): 827-836.
Mohammad M, Trounev A. Fractional nonlinear Volterra–Fredholm integral equations involving Atangana–Baleanu fractional derivative: framelet applications. Adv Differ Equ. 2020; 2020: 1-15. https://doi.org/10.1186/s13662-020-03042-9.
Ahmed A. Existence and Stability Results for Fractional Volterra-Fredholm Integro-Dierential Equation with Mixed Conditions. Adv Dyn Syst Appl. 2021; 16(1): 217-236. https://doi.org/10.37622/adsa/16.1.2021.217-236 .
Sivasankar S, Udhayakumar R. Hilfer Fractional Neutral Stochastic Volterra Integro-Differential Inclusions via Almost Sectorial Operators. Mathematics. 2022; 10(12): 1-19. https://doi.org/10.3390/math10122074
Gebril E, El-Azab MS, Sameeh M. Chebyshev collocation method for fractional Newell-Whitehead-Segel equation. Alex Eng J. 2024; 87: 39-46. https://doi.org/10.1016/j.aej.2023.12.025.
Behera S. Ray SS. An efficient numerical method based on Euler wavelets for solving fractional order pantograph Volterra delay-integro-differential equations. J Comput Appl Math. 2022; 406: 113825. https://doi.org/10.1016/j.cam.2021.113825.
Xu D. Numerical solution of partial integro-differential equation with a weakly singular kernel based on Sinc methods. Math Comput Simul. 2021; 190: 140-158. https://doi.org/10.1016/j.matcom.2021.05.014
Ali I, Yaseen M, Khan S. Addressing Volterra Partial Integro-Differential Equations through an Innovative Extended Cubic B-Spline Collocation Technique. Symmetry 2023; 15(10): 1851. https://doi.org/10.3390/sym15101851
Reddy GMM, Sinha RK, Cuminato JA. A posteriori error analysis of the Crank-Nicolson finite element method for parabolic integro-differential equations. J Sci Comput. 2019; 79: 414–441. https://doi.org/10.1007/s10915-018-0860-1
Rasheed MA, Kadhim SN. Numerical Solutions of Two-Dimensional Vorticity Transport Equation Using Crank-Nicolson Method. Baghdad Sci J. 2022; 19(2): 321-328. https://doi.org/10.21123/bsj.2022.19.2.0321
Al-Rozbayani AM, Al-Botani ZM. Solving Whitham-Broer-Kaup-Like Equations Numerically by using Hybrid Differential Transform Method and Finite Differences Method. Baghdad Sci J. 2022; 19(1): 64-70. https://doi.org/10.21123/bsj.2022.19.1.0064
Noon NJ. Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem. Baghdad Sci J. 2021; 18(4(Suppl.)): 1521-1535. https://doi.org/10.21123/bsj.2021.18.4(Suppl.).1521.
Abd SK, Jari RH. Super convergence of finite element approximations for elliptic problem with Neumann boundary condition. Proceedings of the 1st International Conference on Advanced Research in Pure and Applied Science (ICARPAS 2021): Third Annual Conference of Al-Muthanna University/College of Science, 24–25 March 2021, Al-Samawah, Iraq. AIP Conf. Proc. 2022. 2398(1): 060076. https://doi.org/10.1063/5.0093736.
Pani AK, Sinha RK. Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data. Calcolo. 2000; 37: 181–205. https://doi.org/10.1007/s100920070001.
Reddy GMM, Sinha RK. Ritz–Volterra reconstructions and a posteriori error analysis of finite element method for parabolic integro-differential equations. Equations. IMA J. Numer. Anal. 2015; 35(1): 341–371. https://doi.org/10.1093/imanum/drt059.
Shaw S, Whiteman JR. Numerical Solution of Linear Quasistatic Hereditary Viscoelasticity Problems. SIAM J. Numer. Anal. 2000; 38(1): 80–97. https://doi.org/10.1137/S0036142998337855.
Al-Humedi HA, Al-Abadi AK. Analysis of error estimate for expanded H1 - Galerkin MFEM of PIDEs with nonlinear memory. International Conference on Emergency Applications in Material Science and Technology (ICEAMST 2020), 30–31 January 2020, Namakkal, India. AIP Conf Proc. 2020; 2235(1): 20010. http://dx.doi.org/10.1063/5.0007637.
Yanik EG, Fairweather G. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal Theory Methods Appl. 1988; 12(8): 785–809. https://doi.org/10.1016/0362-546X(88)90039-9.
Kumar L, Sista SG, Sreenadh K. Finite element analysis of parabolic integro-differential equations of Kirchhoff type. Math Methods Appl Sci. 2020; 43: 9129-9150. https://doi.org/10.1002/mma.6607.
Barich F. Some Gronwall–Bellman Inequalities on Time Scales and Their Continuous Forms: A Survey. Symmetry. 2021; 13(2): 198. https://doi.org/10.3390/sym13020198 .
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