It is well known that the complex step method is a tool that calculates derivatives by imposing a complex step in a strict sense. We extended the method by employing the fractional calculus differential operator in this paper. The fractional calculus can be taken in the sense of the Caputo operator, Riemann-Liouville operator, and so forth. Furthermore, we derived several approximations for com...

We have learned that the numerical solution obtained from Euler's method, converges to the exact solution () of the initial value problem ′ = (,), (0) = 0 , as ℎ → 0. We now analyze the convergence of a general one-step method of the form +1 = + ℎΦ(, , ℎ), for some continuous function Φ(, , ℎ). We define the local truncation error of this one-step method by (ℎ) = (+1) − () ℎ − Φ(, (), ℎ). That ...

The Lax-Friedrichs (LxF) method [2, 3, 4] is a basic method for the solution of hyperbolic partial differential equations (PDEs). Its use is limited because its order is only one, but it is easy to program, applicable to general PDEs, and has good qualitative properties because it is monotone. The LxF method is often used to show the effects of dissipation, but it is not actually a dissipative ...

We study excited spherical branes (“giant gravitons”) in AdS × S spacetimes with background flux. For large excitation, these branes may be treated semiclassically. We compute their spectra using Bohr-Sommerfeld quantization and use the AdS/CFT correspondence to relate them to anomalous dimensions in the dual field theory at strong coupling, expressed as a series expansion in powers of 1/N . Th...

We search for nonlinear effects in nuclear giant resonances (GRs), in particular the isovector dipole and the isoscalar quadrupole modes. To that end, we employ a spectral analysis of time-dependent Hartree-Fock (TDHF) dynamics using Skyrme forces. Based on TDHF calculations over a wide range of excitation amplitudes, we explore the collectivity and degree of harmonic motion in these modes. Bot...

We review the step-by-step method of constructing finitely generated free modal algebras. First we discuss the global step-by-step method, which works well for rank one modal logics. Next we refine the global step-by-step method to obtain the local step-by-step method, which is applicable beyond rank one modal logics. In particular, we show that it works well for constructing the finitely gener...