The Coercive Property and a Priori Error Estimation of the Finite Element Method for Linearly Distributed Time Order Fractional Telegraph Equation with Restricted Initial Conditions — Oak Academic Publishing
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The Coercive Property and a Priori Error Estimation of the Finite Element Method for Linearly Distributed Time Order Fractional Telegraph Equation with Restricted Initial Conditions
Department of Mathematics, Delta State University, Abraka, Nigeria
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Department of Mathematics, Delta State University, Abraka, Nigeria
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Department of Mathematics, Dennis Osadebay University, Asaba, Nigeria
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Department of Mathematics, Delta State University, Abraka, Nigeria
,
Department of Mathematics, Delta State University, Abraka, Nigeria
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Department of Mathematics, Delta State University, Abraka, Nigeria
1 Department of Mathematics, Delta State University, Abraka, Nigeria
2 Department of Mathematics, Delta State University, Abraka, Nigeria
3 Department of Mathematics, Dennis Osadebay University, Asaba, Nigeria
4 Department of Mathematics, Delta State University, Abraka, Nigeria
5 Department of Mathematics, Delta State University, Abraka, Nigeria
6 Department of Mathematics, Delta State University, Abraka, Nigeria
Finite Element Method (FEM), when applied to solve problems, has faced some challenges over the years, such as time consumption and the complexity of assumptions. In particular, the making of assumptions has had a significant influence on the accuracy of the method, making it mandatory to carry out sensitivity analysis. The sensitivity analysis helps to identify the level of impact the assumptions have on the method. However, sensitivity analysis via FEM can be very challenging. A priori error estimation, an integral part of FEM, is a basic mathematical tool for predicting the accuracy of numerical solutions. By understanding the relationship between the mesh size, the order of basis functions, and the resulting error, practitioners can effectively design and apply FEM to solve complex Partial Differential Equations (PDEs) with confidence in the reliability of their results. Thus, the coercive property and A priori error estimation based on the L1 formula on a mesh in time and the Mamadu-Njoseh basis functions in space are investigated for a linearly distributed time-order fractional telegraph equation with restricted initial conditions. For this purpose, we constructed a mathematical proof of the coercive property for the fully discretized scheme. Also, we stated and proved a cardinal theorem for a priori error estimation of the approximate solution for the fully discretized scheme. We noticed the role of the restricted initial conditions imposed on the solution in the analysis of a priori error estimation.
KeywordsCoercivityFinite Element MethodMamadu-Njoseh PolynomialsA Priori Error EstimationCauchy-Schwarz InequalityMean Value Theorem
Podlubny, I. (1998) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of Their Solution and Some of Their Applications. Elsevier.
Millar, K.S. and Ross, B. (1999) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons.
Oldham, K.B. and Spanier, J. (1994) Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order. Academic Press.
Hilfer, R. (2000) Application of Fractional Calculus in Physics. World Scientific, 88-99.
Tomasi, W. (2004) Electronic Communication Systems. Prentice Hall.
Cascaval, R.C., Eckstein, E.C., Frota, C.L. and Goldstein, J.A. (2002) Fractional Telegraph Equations. Journal of Mathematical Analysis and Applications , 276, 145-159. https://doi.org/10.1016/s0022-247x(02)00394-3
Mamadu, E.J., Ojarikre, H.I., Ogumeyo, S.A., Iweobodo, D.C., Mamadu, E.A., Tsetimi, J., et al . (2024) A Least Squares Finite Element Method for Time Fractional Telegraph Equation with Vieta-Lucas Basis Functions. Scientific African , 24, e02170. https://doi.org/10.1016/j.sciaf.2024.e02170
Caputo, M. and Mainardi, F. (1971) A New Dissipation Model Based on Memory Mechanism. pure and applied geophysics , 91, 134-147. https://doi.org/10.1007/bf00879562
Yisa, B.M. and Adelabu, N.A. (2022) Numerical Solution of Fractional Order Partial Differential Equation with Sturm-Liouville Problem using Homotopy Peryrubation and Homotopy Analysis Methods. Daffodil International University Journal of Science and Technology , 17, 1-10, 200.
Jafari, H. and Daftardar-Gejji, V. (2006) Solving Linear and Nonlinear Fractional Diffusion and Wave Equations by Adomian Decomposition. Applied Mathematics and Computation , 180, 488-497. https://doi.org/10.1016/j.amc.2005.12.031
Huang, C., An, N. and Yu, X. (2020) A Local Discontinuous Galerkin Method for Time-Fractional Diffusion Equation with Discontinuous Coefficient. Applied Numerical Mathematics , 151, 367-379. https://doi.org/10.1016/j.apnum.2019.11.003
Jia, J. and Wang, H. (2019) A Fast Finite Volume Method for Conservative Space–time Fractional Diffusion Equations Discretized on Space-Time Locally Refined Meshes. Computers & Mathematics with Applications , 78, 1345-1356. https://doi.org/10.1016/j.camwa.2019.04.003
Mamadu, E.J., Ojarikre, H.I. and Maduku, E.O. (2023) Finite Element Orthogonal Collocation Approach for Time Fractional Telegraph Equation with Mamadu-Njoseh Polynomials. Journal of Applied Mathematics and Physics , 11, 2585-2596. https://doi.org/10.4236/jamp.2023.119168
Biazar, J. and Eslami, M. (2010) Analytic Solution for Telegraph Equation by Differential Transform Method. Physics Letters A , 374, 2904-2906. https://doi.org/10.1016/j.physleta.2010.05.012
Chen, J., Liu, F. and Anh, V. (2008) Analytical Solution for the Time-Fractional Telegraph Equation by the Method of Separating Variables. Journal of Mathematical Analysis and Applications , 338, 1364-1377. https://doi.org/10.1016/j.jmaa.2007.06.023
Orsingher, E. and Beghin, L. (2003) Time-fractional Telegraph Equations and Telegraph Processes with Brownian Time. Probability Theory and Related Fields , 128, 141-160. https://doi.org/10.1007/s00440-003-0309-8
Momani, S. (2005) Analytic and Approximate Solutions of the Space-And Time-Fractional Telegraph Equations. Applied Mathematics and Computation , 170, 1126-1134. https://doi.org/10.1016/j.amc.2005.01.009
Huang, J., Cen, Z. and Xu, A. (2022) An Efficient Numerical Method for a Time-Fractional Telegraph Equation. Mathematical Biosciences and Engineering , 19, 4672-4689. https://doi.org/10.3934/mbe.2022217
Tasbozan, O. and Esen, A. (2019) Collocation Solutions for the Time Fractional Telegraph Equation Using Cubic B-Spline Finite Elements. Annals of West University of Timisoara — Mathematics and Computer Science , 57, 131-144. https://doi.org/10.2478/awutm-2019-0020
Ahmed, M., Bhat, A.A. and Jain, R. (2018) Time-Fractional Telegraph Equation and Its Application by Using Adomian Decomposition Method. Journal of New Theory , 2, 73-81.
Solin, P., Segeth, K. and Dolezel, I. (2003) Higher-Order Finite Element Methods. Chapman & Hall/CRC Press.
Logan, D.L. (2011) A First Course in the Finite Element Method. Cengage Learning.
Mamadu, E.J., Ojarikre, H.I. and Njoseh, I.N. (2023) Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation. Mathematics and Statistics , 11, 245-251. https://doi.org/10.13189/ms.2023.110202
Li, Z. and Reed, M.B. (1995) Convergence Analysis for an Element-By-Element Finite Element Method. Computer Methods in Applied Mechanics and Engineering , 123, 33-42. https://doi.org/10.1016/0045-7825(94)00759-g
Wan, X., Yu, H. and Zhai, J. (2018) Convergence Analysis of a Finite Element Approximation of Minimum Action Methods. SIAM Journal on Numerical Analysis , 56, 1597-1620. https://doi.org/10.1137/17m1141679
Mamadu, E.J., Ojarikre, H.I. and Njoseh, I.N. (2023) An Error Analysis of Implicit Finite Difference Method with Mamadu-Njoseh Basis Functions for Time Fractional Telegraph Equation. Asian Research Journal of Mathematics , 19, 20-30. https://doi.org/10.9734/arjom/2023/v19i7675
Tang, Y. (2021) Convergence and Superconvergence of Fully Discrete Finite Element for Time Fractional Optimal Control Problems. American Journal of Computational Mathematics , 11, 53-63. https://doi.org/10.4236/ajcm.2021.111005
Wang, H., Yang, D. and Zhu, S. (2016) Accuracy of Finite Element Methods for Boundary-Value Problems of Steady-State Fractional Diffusion Equations. Journal of Scientific Computing , 70, 429-449. https://doi.org/10.1007/s10915-016-0196-7
Hao, Z., Park, M., Lin, G. and Cai, Z. (2018) Finite Element Method for Two-Sided Fractional Differential Equations with Variable Coefficients: Galerkin Approach. Journal of Scientific Computing , 79, 700-717. https://doi.org/10.1007/s10915-018-0869-5
Njoseh, I.N. and Mamadu, E.J. (2016) Numerical Solutions of Fifth Order Boundary Value Problems Using Mamadu-Njoseh Polynomials. Science World Journals , 11, 21-24.