Using Maple to Study the Partial Differential Problems

This paper uses the mathematical software Maple for the auxiliary tool to study the partial differential problem of four types of multivariable functions.We can obtain the infinite series forms of any order partial derivatives of these four types of multivariable functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values.On the other hand, we propose some examples to do calculation practically.The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple.This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations.For this reason, Maple provides insights and guidance regarding problem-solving methods.

This article takes the mathematical software Maple as the auxiliary tool to study the partial differential problems of four types of two-variables functions. We can obtain the infinite series forms of any order partial derivatives of these two-variables functions by using differentiation term by term theorem. In addition, we propose some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying the answers by using Maple.

Recovery of a manifold with boundary and its continuity as a function of its metric tensor

A cubic Hermite finite element-continuation method for numerical solutions of the von Kármán equations

On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values

The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary. In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results will also be presented.

This paper presents the calculation of the classic-curvature and the intensity-curvature term before interpolation of a bivariate polynomial model function. The classic-curvature is termed as yc (x, y) and the intensity-curvature term before interpolation is termed as E0. The classic-curvature is defined as the sum of the four second order partial derivatives of the bivariate polynomial. The intensity-curvature term before interpolation is defined as the integral of the product between the pixel intensity value termed as f(0, 0) and the classic-curvature calculated at the origin of the coordinate system of the pixel. This paper presents an application of the calculation of classic-curvature and the intensity- curvature term before interpolation using two- dimensional Magnetic Resonance Imaging (MRI) data and reports for the first time in the literature on the behavior of the intensity-curvature term before interpolation. Index Terms—Magnetic Resonance Imaging (MRI), classic-curvature, intensity-curvature term before interpolation, intensity-curvature measurement approaches, first order partial derivative, second order partial derivative, bivariate polynomial, model function, image.

A generalized compound model (the GC-PDF) for radar clutter is proposed. The speckle of the radar echo and its mean are described using the generalized gamma distribution. The GC-PDF is expressed either in integral or infinite series form. However, its moments are derived in closed form so that they can be used in clutter classification. The proposed model comprises the K-distribution as a special case. The generalized K-PDF is also defined. >

New, exact expressions for the probability density and distribution functions of a bivariate Hoyt (Nakagami-q) process with arbitrary correlation pattern in a nonstationary environment are derived, a solution to a longstanding unsolved problem. More specifically, the following are obtained: joint probability density function, joint cumulative distribution function, power and envelope correlation coefficients, and some statistics related to the signal-to-noise ratio at the output of the selection combiner, namely, outage probability and probability density function. The exact expressions are given in infinite series form, but are mathematically tractable, easy to evaluate, and flexible enough to accommodate a myriad of correlation scenarios, useful in the analysis of a more general fading environment. The power correlation coefficient is found in an exact, simple, closed-form formula, whereas the envelope one, also exact, is more mathematically elaborate. Approximate, simple closed-form expressions for the joint distribution are also obtained that yield (i) excellent results for small to medium correlation coefficients and (ii) reasonable ones for high correlation coefficient.

We prove that a generic homogeneous polynomial of degree $d$ is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order $k$ for $k\leqslant \frac{d}{2}-1$.

Complexity of solving nonlinear equations in the deterministic, randomized and quantum settings

Linear dependence of a function set of m variables with vanishing generalized Wronskians

This work is motivated by the investigation of a fractional extension of a general nonlinear multidimensional wave equation with damping. The model under study considers partial derivatives of orders in $$(0, 1) \cup (1, 2]$$ with respect to the spatial variables. The undamped one-dimensional version of the model was previously used in the literature to investigate the presence of the nonlinear phenomenon of supratransmission in fractional relativistic wave equations. In agreement with the continuous counterpart, the capability of the method to preserve the total energy of the system was demonstrated then. In the present work, we show that the modified methodology is capable of preserving the dissipation of energy of the extended continuous model. We note that the discrete energy functional is positive, a fact which is in agreement with the positivity of the continuous energy. Moreover, the unique solubility of the method is established, and we show that our technique is second-order consistent, stable and quadratically convergent. Some bounds for the numerical solution are computed in the way, and various simulations are performed in order to assess the capability of the method to preserve or dissipate the energy of the system.

A note on Bartholdi zeta function and graph invariants based on resistance distance

Bonds Futures: Delta? No Gamma!