D. S. Zhang

SHANNON-GABOR WAVELET DISTRIBUTED APPROXIMATING FUNCTIONAL

Abstract The Shannon sampling theorem is critically reviewed from a physical point of view. An approximate sampling formula is proposed, combining Shannon sampling with a Gabor-distributed approximating functional (DAF) window function, which results in new Shannon–Gabor wavelet DAFs (SGWDs). They are extremely smooth, decay rapidly, have simultaneous time-frequency localization, and are also generalized delta sequences (reducing to the Dirac delta function under the limit of a zero window width). Shannons sampling theorem is recovered exactly when the window is infinitely wide.Finally, SGWDs are well-behaved L 2 ( R ) kernels, and thus can be used for solving differential equations.

Burgers’ equation with high Reynolds number

Burgers’ equation, involving very high Reynolds numbers, is numerically solved by using a new approach based on the distributed approximating functional for representing spatial derivatives of the velocity field. For moderately large Reynolds numbers, this simple approach can provide very high accuracy while using a small number of grid points. In the case where the Reynolds number ⩾105, the exact solution is not available and a discrepancy exists in the literature. Our results clarify the behavior of the solution under these conditions.

Wavelet-distributed approximating functional method for solving the Navier-Stokes equation

The Navier-Stokes equations with both periodic and non-slip boundary conditions are solved using a new class of wavelets based on distributed approximating functionals (DAFs). Extremely high accuracy is found in our wavelet-DAF integration of the analytically solvable Taylor problem, using 32 grid points in each of the two spatial dimensions, for Reynolds numbers from Re = 20 to Re = ∞. The present approach is then applied to the lid-driven cavity problem with standard non-slip boundary conditions. Physically reasonable solutions are obtained for Reynolds numbers as high as 3200, using 63 grid points in each spatial dimension. Our results indicate that wavelet methods are readily applicable to those dynamical problems for which the existence of possible singularities demands highly accurate solution methods.

Distributed approximating functional approach to the Fokker–Planck equation: Eigenfunction expansion

The distributed approximating functional method is applied to the solution of the Fokker–Planck equations. The present approach is limited to the standard eigenfunction expansion method. Three typical examples, a Lorentz Fokker–Planck equation, a bistable diffusion model and a Henon–Heiles two-dimensional anharmonic resonating system, are considered in the present numerical testing. All results are in excellent agreement with those of established methods in the field. It is found that the distributed approximating functional method yields the accuracy of a spectral method but with a local method’s simplicity and flexibility for the eigenvalue problems arising from the Fokker–Planck equations.

Distributed approximating functional approach to the Fokker–Planck equation: Time propagation

The Fokker–Planck equation is solved by the method of distributed approximating functionals via forward time propagation. Numerical schemes involving higher-order terms in Δt are discussed for the time discretization. Three typical examples (a Wiener process, an Ornstein–Uhlenbeck process, and a bistable diffusion model) are used to test the accuracy and reliability of the present approach, which provides solutions that are accurate up to ten significant figures while using a small number of grid points and a reasonably large time increment. Two sets of solutions for the bistable system, one computed using the eigenfunction expansion of a preceding paper and the other using the present time-dependent treatment, agree to no fewer than five significant figures. It is found that the distributed approximating functional method, while simple in its implementation, yields the most accurate numerical solutions yet available for the Fokker–Planck equation.

Lagrange distributed approximating functional method for the solution of the Schrödinger equation

Abstract The utility and accuracy of a new distributed approximating functional (DAF), combining the Gaussian weighted DAF concept with Lagrange interpolation, is explored for the discrete spectrum solution of the Schrodinger equation. Two instructive examples, an I 2 Morse oscillator and the 2-dimensional Henon–Heiles potential, are considered in the present study. The present “Lagrange DAF” (LDAF) approach achieves extremely high accuracy for I 2 while using fewer grid points than previous approaches. The present results for the Henon–Heiles system are in excellent agreement with those of earlier established methods, such as that of Shizgal.

Distributed approximating functional treatment of noisy signals

Abstract Based on their so-called “well-tempered” property, distributed approximating functionals (DAFs) are shown to be very good data filters, and consequently, they have many potential applications in all fields of engineering and the natural sciences. In this paper, a periodic extension of a noisy signal is proposed to generate a “pseudo-signal” in the infinite domain, enabling the use of noncausal, zero-phase window filters that require a knowledge of the signal in the extended domain. The extended signal is also useful for the application of fast Fourier transforms (in which the preferred number of sampled data points is a power of 2). The most attractive feature of the method is that it introduces little aliasing between the original and true signal. The resulting extended signal is then filtered by using DAFs as low pass filters under the assumption that the true signal is bandwidth limited and most of the noise components are in the high frequency region. A “signature” based on computing the root-mean-square value of the filtered signal is introduced to indicate when the high frequency noise has been eliminated with the choice of the DAF parameters. To illustrate the usefulness of the present algorithm, two noisy signal examples are periodically extended and filtered using DAFs.