A numerical approach for a class of time-fractional reaction-diffusion equation through exponential B-spline method is presented in this paper. The proposed scheme is a combination of Crank-Nicolson method for the Caputo time derivative and exponential B-spline method for space derivative. The unconditional stability and convergence of the proposed scheme are presented. Several numerical examples are presented to illustrate the feasibility and efficiency of the proposed scheme.

Modified Christoffel equations are derived for three-dimensional wave propagation in a general anisotropic medium under initial stress. The three roots of a cubic equation define the phase velocities of three quasi-waves in the medium. Analytical expressions are used to calculate the directional derivatives of phase velocities. These derivatives are, further, used to calculate the group velocities and ray directions of the three quasi-waves in a pre-stressed anisotropic medium. Effect of initial stress on wave propagation is observed through the deviations in phase velocity, group velocity and ray direction for each of the quasi-waves. The variations of these deviations with the phase direction are plotted for a numerical model of general anisotropic medium with triclinic/ monoclinic/orthorhombic symmetry.

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This paper focuses on the numerical solution of the variable coefficient multiterm time fractional advection-diffusion equation via exponential B-splines. We discretize the temporal part by using the Crank-Nicolson method and spatial part by the exponential B-splines. The unconditional stability is obtained by the Von-Neumann method. The convergence rates are also studied. Numerical simulations confirm the theoretically expected accuracy in both time and space directions. A comparative analysis with the other methods shows the superiority of the proposed algorithm.