A course in mathematical analysis.

An unabridged and unaltered republication of the Hedrick-Dunkel translation (v. 1-2); v. 3. newly translated by Howard G. Bergmann.

Elementary differential equations /

Mode of access: Internet.

Nonlinear differential equations and nonlinear oscillations : final report /

At head of title: Office of Naval Research, Contract NONR-1858(04), Project NRO43-942.

Differential equations applied in science and engineering

Mode of access: Internet.

First principles of the differential and integral calculus, or, The doctrines of fluxions ... for the use of the students of the university at Cambridge, New England : taken chiefly from the mathematics of Bézout.

Mode of access: Internet.

A treatise on differential equations

Mode of access: Internet.

A treatise on differential equations,

Mode of access: Internet.

Non-linear analysis of the thermo-electro-mechanical behaviour of shear deformable FGM plates with piezoelectric actuators

This paper investigates the non-linear bending behaviour of functionally graded plates that are bonded with piezoelectric actuator layers and subjected to transverse loads and a temperature gradient based on Reddy's higher-order shear deformation plate theory. The von Karman-type geometric non-linearity, piezoelectric and thermal effects are included in mathematical formulations. The temperature change is due to a steady-state heat conduction through the plate thickness. The material properties are assumed to be graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The plate is clamped at two opposite edges, while the remaining edges can be free, simply supported or clamped. Differential quadrature approximation in the X-axis is employed to convert the partial differential governing equations and the associated boundary conditions into a set of ordinary differential equations. By choosing the appropriate functions as the displacement and stress functions on each nodal line and then applying the Galerkin procedure, a system of non-linear algebraic equations is obtained, from which the non-linear bending response of the plate is determined through a Picard iteration scheme. Numerical results for zirconia/aluminium rectangular plates are given in dimensionless graphical form. The effects of the applied actuator voltage, the volume fraction exponent, the temperature gradient, as well as the characteristics of the boundary conditions are also studied in detail. Copyright (C) 2004 John Wiley Sons, Ltd.

Implicit stochastic Runge-Kutta methods for stochastic differential equations

In this paper we construct implicit stochastic Runge-Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods.

Treatment of partial-thickness burns: A prospective, randomized trial using Transcyte (TM)

Background: The purpose of the present study was to compare the effectiveness of three burns dressings (TransCyte, a bio-engineered skin substitute; Biobrane; and Silvazine cream (silver sulphadiazine and 0.2% chlorhexidine)), in treating children with partial-thickness burns. The primary objective was to determine the days until greater than or equal to90% re-epithelialization. The secondary objectives were to evaluate the number of wounds requiring autografting and the number of dressing changes/local wound care required. Methods: Study wounds were identified on each patient and the patients were randomized to receive TransCyte or Biobrane or Silvazine. Assessment of study wound closure began at 2 days after treatment and continued at least every other day thereafter until the wounds re-epithelialized or were autografted. A laser Doppler imaging system was used as an adjunct to assessing the depth of the burn. Results: Thirty-three patients with 58 wound sites enrolled in the study (TransCyte, n = 20, Biobrane, n = 17; Silvazine, n = 21). Mean time to re-epithelialization was 7.5 days for TransCyte, 9.5 days for Biobrane, and 11.2 days for Silvazine. The number of wounds requiring autografting were 5/21 (24%) for Silvazine, 3/17 (17%) for Biobrane, and 1/20 (5%) for TransCyte. Conclusions: When used in partial-thickness burns in children, TransCyte promotes fastest re-epithelialization and required less overall dressings then Biobrane or Silvazine. Patients who received Silvazine or Biobrane require more autografting than those treated with TransCyte.

Analytical solutions of linear finite- and small-strain one-dimensional consolidation

Analytical solutions are presented for linear finite-strain one-dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one- and two-way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small-strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small-strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non-linear finite-strain consolidation. They also clarify a formerly perplexing aspect of finite-strain solution charts first noted in numerical solutions. Copyright (C) 2004 John Wiley Sons, Ltd.

Advances in distributed parameter approach to the dynamics and control of activated sludge processes for wastewater treatment

This paper presents a review of modelling and control of biological nutrient removal (BNR)-activated sludge processes for wastewater treatment using distributed parameter models described by partial differential equations (PDE). Numerical methods for solution to the BNR-activated sludge process dynamics are reviewed and these include method of lines, global orthogonal collocation and orthogonal collocation on finite elements. Fundamental techniques and conceptual advances of the distributed parameter approach to the dynamics and control of activated sludge processes are briefly described. A critical analysis on the advantages of the distributed parameter approach over the conventional modelling strategy in this paper shows that the activated sludge process is more adequately described by the former and the method is recommended for application to the wastewater industry (c) 2006 Elsevier Ltd. All rights reserved.

Geodesics without differential equations: general relativistic calculations for introductory modern physics classes

Introductory courses covering modem physics sometimes introduce some elementary ideas from general relativity, though the idea of a geodesic is generally limited to shortest Euclidean length on a curved surface of two spatial dimensions rather than extremal aging in spacetime. It is shown that Epstein charts provide a simple geometric picture of geodesics in one space and one time dimension and that for a hypothetical uniform gravitational field, geodesics are straight lines on a planar diagram. This means that the properties of geodesics in a uniform field can be calculated with only a knowledge of elementary geometry and trigonometry, thus making the calculation of some basic results of general relativity accessible to students even in an algebra-based survey course on physics.

Stochastic delay differential equations for genetic regulatory networks

Time delay is an important aspect in the modelling of genetic regulation due to slow biochemical reactions such as gene transcription and translation, and protein diffusion between the cytosol and nucleus. In this paper we introduce a general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks. Based on recent developments with the delay stochastic simulation algorithm, the delay chemical masterequation and the delay reaction rate equation are developed for describing biological reactions with time delay, which leads to stochastic delay differential equations derived from the Langevin approach. Two simple genetic regulatory networks are used to study the impact of' intrinsic noise on the system dynamics where there are delays. (c) 2006 Elsevier B.V. All rights reserved.

Student Difficulties with units in differential equations in modelling contexts

First-year undergraduate engineering students' understanding of the units of factors and terms in first-order ordinary differential equations used in modelling contexts was investigated using diagnostic quiz questions. Few students appeared to realize that the units of each term in such equations must be the same, or if they did, nevertheless failed to apply that knowledge when needed. In addition, few students were able to determine the units of a proportionality factor in a simple equation. These results indicate that lecturers of modelling courses cannot take this foundational knowledge for granted and should explicitly include it in instruction.