Finite Element Method and numerical study for a super-linear reaction-diffusion problem with integral conditions

Sihem Benamira; Takieddine Oussaeif; Sofiane dehilis; Abdelfatah bouziani

doi:10.55059/ijm.2022.1.1/11

Authors

Sihem Benamira
Takieddine Oussaeif
Sofiane dehilis https://orcid.org/0000-0002-8771-5046
Abdelfatah bouziani

DOI:

https://doi.org/10.55059/ijm.2022.1.1/11

Keywords:

Non-local conditions, finite element method, A priori estimates, super-linear parabolic equation., numerical study

Abstract

In this work, we prove the existence, uniqueness, and continuous dependence of generalized solution of a nonlinear reaction-diffusion problem with only integral terms in the boundaries, by using the finite element method.Also we have developed an efficient numerical finite difference schemes. Some numerical results are reported to show the efficiency and accuracy of the scheme.

References

A.Bouziani, On a class of nonlinear reaction-Diffusion systems with nonlocal boundary conditions, A analysis 2004. 9 (2004), 793-813. https://doi.org/10.1155/S1085337504311061

A.Bouziani, N.Merazga, S.Benamira, Galerkin method applied to a parabolic evolution problem with nonlocalboundary conditions . Non linear Analysis. 69 (2008), 1515-1524. https://doi.org/10.1016/j.na.2007.07.008

A.Bouziani, Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl.Math. Stochatic. Anal. 9 (1996), 323-330. https://doi.org/10.1155/S1048953396000305

A. Bouziani, N.Merazga, On a time-discretisation method for a semilinear heat equation with purely integralcondition in a nonclassical function . Non linear analysis. 66 ( 2007), 604-623. https://doi.org/10.1016/j.na.2005.12.005

M.Dehghan, Fully explicit finite-differenc emethod for two-dimensional diffusion with an integral condition,nonlinear Anal. 48 (2002), no. 5, 637-650. https://doi.org/10.1016/S0362-546X(00)00172-3

C.V.Pao, Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions,J.Comput.Appl.Math.136 (2001), no.1-2, 227-243. https://doi.org/10.1016/S0377-0427(00)00614-2

T. E Oussaeif, A Bouziani, Solvability of nonlinear viscosity Equation with a boundary integral condition, J.Nonlinear Evol. Equ. Appl. 3 (2015), 31-45.

T. E Oussaeif, A Bouziani; Solvability of Nonlinear Goursat Type Problem for Hyperbolic Equation withIntegral Condition; Khayyam Journal of Mathematics. 4, no. 2, 198-213.

Bouziani A, T-E Oussaeif and L. Benaoua, "A Mixed Problem with an Integral Two-Space-Variables Con-dition for Parabolic Equation with The Bessel Operator", Journal of Mathematics. 2013 (2013), Article ID 457631, 8 pages. https://doi.org/10.1155/2013/457631

Dhelis S, Bouziani A, Oussaeif T-E, Study of Solution for a Parabolic Integrodifferential Equation with theSecond Kind Integral Condition, Int. J. Anal. Appl. 16 (2018), no. 4, 569-593.

O. Taki-Eddine, B Abdelfatah; A priori estimates for weak solution for a time-fractional nonlinear reaction-diffusion equations with an integral condition,Chaos, Solitons & Fractals. 103 (2017), 79-89. https://doi.org/10.1016/j.chaos.2017.05.035

Oussaeif T-E, Bouziani A. Existence and uniqueness of solutions to parabolic fractional differential equationswith integral conditions. Electron J. Differ. Equ.179 (2014), 1-10.

T-E Oussaeif, A Bouziani, Mixed Problem with an Integral Two-Space-Variables Condition for a Class ofHyperbolic Equations, International Journal of Analysis. 2013 (2013), 8 pages. https://doi.org/10.1155/2013/957163

T-E Oussaeif, A Bouziani, Mixed Problem with an Integral Two-Space-Variables Condition for a ParabolicEquation. International Journal of Evolution Equations. 9, no. 2, 181-198.

T.-E. Oussaeif, A. Bouziani, Mixed Problem with an Integral Two-Space-Variables condition for a ThirdOrder Parabolic Equation, International Journal of Analysis and Applications. 12 (2016), no. 2, 98-117.

T.-E. Oussaeif, A. Bouziani, Solvability of nonlinear viscosity equation with a boundary integral condition,J. Nonl. Evol. Equ. Appl. 2015 (2015), no. 3, 31-45.21.

T-E Oussaeif, A Bouziani and N Gattal, Numerical Solution of Mixed Problem of Parabolic Equation withan Integral Conditions by using Finite Difference and Orthogonal Function Approximation, MathematicaMoravica. 16 (2012), no. 2, 89-98. https://doi.org/10.5937/MatMor1202089T

Francoise Truc Systemes differentiels non lineaires October 2010.

T.-E. Oussaeif, A. Bouziani, Nonlocal problem for a third order partial differential equation of mixed typewith an integral two-space-variables condition, Communications in Optimization Theory. 2017 (2017), 1-16, Article ID 16. https://doi.org/10.23952/cot.2017.16

B. Sihem, O. Taki Eddine and B. Abdelfatah; Galerkin Finite Element Method for a Semi-linear ParabolicEquation with Purely Integral Conditions; Bol. Soc. Paran. Mat. https://doi.org/10.5269/bspm.44918.

B. Sihem, O. Taki Eddine and B. Abdelfatah, study a semi-linear pseudo-parabolic problem with Neumannand integral conditions by using Galerkin mixed finite element method; Communications in Nonlinear Analysis. 9, Issue 1, 1-20.22.