Abstract
In this paper, we study the Caputo–Hadamard fractional partial differential equation where the time derivative is the Caputo–Hadamard fractional derivative and the space derivative is the integer-order one. We first introduce a modified Laplace transform. Then using the newly defined Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution to this kind of linear equation. Furthermore, we study the regularity and logarithmic decay of its solution. Since the equation has a time fractional derivative, its solution behaves a certain weak regularity at the initial time. We use the finite difference scheme on non-uniform meshes to approximate the time fractional derivative in order to guarantee the accuracy and use the local discontinuous Galerkin method (LDG) to approximate the spacial derivative. The fully discrete scheme is established and analyzed. A numerical example is displayed which support the theoretical analysis.
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References
Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-Type Fractional Differential Equations. Springer, Switzerland (2017)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amserdam (1978)
Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection–diffusion problems. Math. Comput. 71, 455–478 (2002)
Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
Debnath, L., Bhatta, D.: Integral Transforms and Their Applications. Chapman and Hall/CRC, Boca Raton (2007)
Du, Y.W., Liu, Y., Li, H., Fang, Z.C., He, S.: Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation. J. Comput. Phys. 344, 108–126 (2017)
Gohar, M., Li, C.P., Yin, C.T.: On Caputo–Hadamard fractional differential equations. Int. J. Comput. Math. 97, 1459–1483 (2020)
Gohar, M., Li, C.P., Li, Z.Q.: Finite difference methods for Caputo-Hadamard fractional differential equations. Mediterr. J. Math. 17, 194 (2020)
Garra, R., Mainardi, F., Spada, G.: A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos Solitons Fractals 102, 333–338 (2017)
Hadamard, J.: Essai sur létude des fonctions données par leur développement de Taylor. J. Math. Pures Appl. 8, 101–186 (1892)
Jarad, F., Abdeljawad, T.: Generalized fractional derivatives and Laplace transform. Discrete Contin. Dyn. S 13, 709–722 (2020)
Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012)
Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38, 1191–1204 (2001)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Li, C.P., Cai, M.: Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia (2019)
Li, C.P., Wang, Z.: The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: numerical analysis. Appl. Numer. Math. 140, 1–22 (2019)
Li, C.P., Wang, Z.: The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: mathematical analysis. Appl. Numer. Math. 150, 587–606 (2020)
Li, C.P., Wang, Z.: The discontinuous Galerkin finite element method for Caputo-type nonlinear conservation law. Math. Comput. Simulat. 169, 51–73 (2020)
Li, C.P., Li, Z.Q.: Asymptotic behaviors of solution to Caputo–Hadamard fractional partial differential equation with fractional Laplacian. Int. J. Comput. Math. (2020). https://doi.org/10.1080/00207160.2020.1744574
Luchko, Y.: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351, 218–223 (2009)
Luchko, Y.: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15, 141–160 (2012)
Liu, Y., Yan, Y., Khan, M.: Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equations. Appl. Numer. Math. 115, 200–213 (2017)
Ma, L.: On the kinetics of Hadamard-type fractional defferential systems. Fract. Calc. Appl. Anal. 23, 553–570 (2020)
Ma, L., Li, C.P.: On Hadamard fractional calculus. Fractals 25, 1750033 (2017)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)
Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)
Wei, L.L., He, Y.N.: Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl. Math. Model. 38, 1511–1522 (2014)
Xu, Q.W., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52, 405–423 (2014)
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The work was partially supported by the National Natural Science Foundation of China under Grant No. 11872234.
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Li, C., Li, Z. & Wang, Z. Mathematical Analysis and the Local Discontinuous Galerkin Method for Caputo–Hadamard Fractional Partial Differential Equation. J Sci Comput 85, 41 (2020). https://doi.org/10.1007/s10915-020-01353-3
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DOI: https://doi.org/10.1007/s10915-020-01353-3
Keywords
- Caputo–Hadamard derivative
- Regularity
- Finite difference scheme on non-uniform meshes
- Local discontinuous Galerkin method
- Stability and convergence