The textbook first encounter with nonlinearity in a partial differential equation (PDE) is the first-order wave equation: ut + uux = 0. Often referred to as the inviscid Burgers equation, many are familiar with this equation in the theoretical contexts of characteristics, wavebreaking, or shock propagation. Another canonical behavior contained within this simplest of PDEs is the spectral cascad...

The Fourier optics technique is founded on the transfer function derived from the scalar wave equation and thus has traditionally been applied to monochromatic scalar fields propagating paraxially in isotropic lossless materials. We review scalar and vector diffraction theory to show that the scalar Fourier optics technique can be seamlessly extended to the case of time-dependent nonparaxial an...

In this paper the effect of magnetic field on linear and non-linear thermal instability in an anisotropic porous medium saturated with viscoelastic fluid is considered. Normal mode technique is used to investigate the linear stability analysis, while non-linear stability has been done using minimal representation of truncated Fourier series involving only two terms. Extended Darcy model, which ...

In order to expand a formal linearization method on designing an observer for nonlinear dynamic and measurement systems, we exploit the discrete Fourier expansion that can reduce computational complexity. A nonlinear multidimensional dynamic system is considered by a nonlinear ordinary differential equation, and a measurement equation is done by a nonlinear equation. Defining a linearization fu...

1. Everything is number 2. Ancient Astronomy 3. Optics and statics 4. The middle ages and the renaissance 5. Mechanics of motion 6. Newtonian mechanics 7. Early differential equations 8. The Brachistochrone 9. Early methods of solutionLinear differential equations 10 Newton’s second law as a differential equationThe method of perturbations 11. The vibrating stringPartial differential equations ...

A system of PDOs(=Partial Differential Operators) has two non-commutativities, (i) one from [∂q, q] = 1 (Heisenberg relation), (ii) the other from [A, B] 6= 0 for A, B being matrices in general. Non-commutativity from Heisenberg relation is nicely controlled by using Fourier transformations (i.e. the theory of ΨDOs=pseudo-differential operators). Here, we give a new method of treating non-commu...

Photon density wave is the modulated optical intensity wave which propagates through the diffuse medium and exhibits the amplitude and phase variations. It satisfies the time-dependent radiative transfer equation and is similar to temperature wave field, and has been used in medical imaging. This paper discusses the basic characteristics of the photon density wave and gives exact numerical solu...

In this paper, we develop a novel energy-preserving wavelet collocation method for solving general multi-symplectic formulations of Hamiltonian PDEs. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, the wavelet collocation method is conducted for spatial discretization. The obtained semi-discrete system is shown to be a finite-dimensional Hamiltonian s...

In this paper we consider Runge-Kutta discontinuous Galerkin (RKDG) schemes for Vlasov-Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preservi...

The efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics. Previous wave equations which use time-domain fractional operators require the storage of the complete pressure field at previous time steps (such operators are convolution based). This makes them unsuitable for many th...