Importance of Understanding the Physical System in Selecting Separation of Variables Based Methods to Solve the Heat Conduction Partial Differential Equation

Separation of variables is a common method for producing an analytical based solution to partial differential equations. Despite the wide application of this method, often the physical phenomena described by the differential equations are not adequately involved in the discourse over the appropriate methods to solve a given problem, particularly in mathematics curricula. However, as mathematics is the tool to better understanding of the physical world, the meaning of the differential equation, boundary conditions, and initial conditions cannot be detached from the methods used to solve the differential equations. Failure to recognize the physical conditions being studied can lead to solution methods that are invalid or unphysical. This paper demonstrates how awareness of the physical nature of the system being investigated and its relationship to the mathematics can guide the selection of the relevant solution methods. To illustrate the importance of the comprehension of the physical meaning behind the mathematical equations and representations and the need to avoid rote application a solution technique, the logic behind the selection of the appropriate solution techniques for the one-dimensional transient heat conduction equation is considered under different imposed conditions which lead to different trends in system operation

A Generalized Solution Method to Undamped Constant-Coefficient Second-Order ODEs Using Laplace Transforms and Fourier Series

A generalized method for solving an undamped second order, linear ordinary differential equation with constant coefficients is presented where the non-homogeneous term of the differential equation is represented by Fourier series and a solution is found through Laplace transforms. This method makes use of a particular partial fraction expansion form for finding the inverse Laplace transform. If a non-homogeneous function meets certain criteria for a Fourier series representation, then this technique can be used as a more automated means to solve the differential equation as transforms for specific functions need not be determined. The combined use of the Fourier series and Laplace transforms also reinforces the understanding of function representation through a Fourier series and its potential limitations, the mechanics of finding the Laplace transform of a differential equation and inverse transforms, the operation of an undamped system, and through programming insight into the practical application of both tools including information on the influence of the number of terms in the series solution

Undetermined Coefficients with Hyperbolic Sines and Cosines

The method of undetermined coefficients is commonly applied to solve linear, constant coefficient, non-homogeneous ordinary differential equations when the forcing function is from a selected class of functions. Often the hyperbolic sine and cosine functions are not explicitly included in this list of functions. Through a set of guided examples, this work argues that the hyperbolic sine and cosine ought to be included in the select class of functions. Careful explanation is provided for the necessary treatment of the cases where the argument of the hyperbolic sine and/or cosine functions matches one or both of the roots of the characteristic equation of the differential equation. Finally, a generalized approach where the hyperbolic and trigonometric sine and cosine functions are written in their exponential form illustrates the connections between the exponential, trigonometric, and hyperbolic sine and cosine functions. This exploration leads to a deeper understanding of the method of undetermined coefficients and can be adapted into coursework on the undetermined coefficients topic

Special Case of Partial Fraction Expansion with Laplace Transform Application

Partial fraction expansion is often used with the Laplace Transforms to formulate algebraic expressions for which the inverse Laplace Transform can be easily found. This paper demonstrates a special case for which a linear, constant coefficient, second order ordinary differential equation with no damping term and a harmonic function non-homogeneous term leads to a simplified partial fraction expansion due to the decoupling of the partial fraction expansion coefficients of s and the constant coefficients. Recognizing this special form can allow for quicker calculations and automation of the solution to the differential equation form which is commonly used to model physical systems

Guidelines for the use and interpretation of assays for monitoring autophagy (4th edition)1.

In 2008, we published the first set of guidelines for standardizing research in autophagy. Since then, this topic has received increasing attention, and many scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Thus, it is important to formulate on a regular basis updated guidelines for monitoring autophagy in different organisms. Despite numerous reviews, there continues to be confusion regarding acceptable methods to evaluate autophagy, especially in multicellular eukaryotes. Here, we present a set of guidelines for investigators to select and interpret methods to examine autophagy and related processes, and for reviewers to provide realistic and reasonable critiques of reports that are focused on these processes. These guidelines are not meant to be a dogmatic set of rules, because the appropriateness of any assay largely depends on the question being asked and the system being used. Moreover, no individual assay is perfect for every situation, calling for the use of multiple techniques to properly monitor autophagy in each experimental setting. Finally, several core components of the autophagy machinery have been implicated in distinct autophagic processes (canonical and noncanonical autophagy), implying that genetic approaches to block autophagy should rely on targeting two or more autophagy-related genes that ideally participate in distinct steps of the pathway. Along similar lines, because multiple proteins involved in autophagy also regulate other cellular pathways including apoptosis, not all of them can be used as a specific marker for bona fide autophagic responses. Here, we critically discuss current methods of assessing autophagy and the information they can, or cannot, provide. Our ultimate goal is to encourage intellectual and technical innovation in the field

Guidelines for the use and interpretation of assays for monitoring autophagy (3rd edition)

In 2008 we published the first set of guidelines for standardizing research in autophagy. Since then, research on this topic has continued to accelerate, and many new scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Accordingly, it is important to update these guidelines for monitoring autophagy in different organisms. Various reviews have described the range of assays that have been used for this purpose. Nevertheless, there continues to be confusion regarding acceptable methods to measure autophagy, especially in multicellular eukaryotes. For example, a key point that needs to be emphasized is that there is a difference between measurements that monitor the numbers or volume of autophagic elements (e.g., autophagosomes or autolysosomes) at any stage of the autophagic process versus those that measure fl ux through the autophagy pathway (i.e., the complete process including the amount and rate of cargo sequestered and degraded). In particular, a block in macroautophagy that results in autophagosome accumulation must be differentiated from stimuli that increase autophagic activity, defi ned as increased autophagy induction coupled with increased delivery to, and degradation within, lysosomes (inmost higher eukaryotes and some protists such as Dictyostelium ) or the vacuole (in plants and fungi). In other words, it is especially important that investigators new to the fi eld understand that the appearance of more autophagosomes does not necessarily equate with more autophagy. In fact, in many cases, autophagosomes accumulate because of a block in trafficking to lysosomes without a concomitant change in autophagosome biogenesis, whereas an increase in autolysosomes may reflect a reduction in degradative activity. It is worth emphasizing here that lysosomal digestion is a stage of autophagy and evaluating its competence is a crucial part of the evaluation of autophagic flux, or complete autophagy. Here, we present a set of guidelines for the selection and interpretation of methods for use by investigators who aim to examine macroautophagy and related processes, as well as for reviewers who need to provide realistic and reasonable critiques of papers that are focused on these processes. These guidelines are not meant to be a formulaic set of rules, because the appropriate assays depend in part on the question being asked and the system being used. In addition, we emphasize that no individual assay is guaranteed to be the most appropriate one in every situation, and we strongly recommend the use of multiple assays to monitor autophagy. Along these lines, because of the potential for pleiotropic effects due to blocking autophagy through genetic manipulation it is imperative to delete or knock down more than one autophagy-related gene. In addition, some individual Atg proteins, or groups of proteins, are involved in other cellular pathways so not all Atg proteins can be used as a specific marker for an autophagic process. In these guidelines, we consider these various methods of assessing autophagy and what information can, or cannot, be obtained from them. Finally, by discussing the merits and limits of particular autophagy assays, we hope to encourage technical innovation in the field

Guidelines for the use and interpretation of assays for monitoring autophagy (4th edition)

International audienceIn 2008, we published the first set of guidelines for standardizing research in autophagy. Since then, this topic has received increasing attention, and many scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Thus, it is important to formulate on a regular basis updated guidelines for monitoring autophagy in different organisms. Despite numerous reviews, there continues to be confusion regarding acceptable methods to evaluate autophagy, especially in multicellular eukaryotes. Here, we present a set of guidelines for investigators to select and interpret methods to examine autophagy and related processes, and for reviewers to provide realistic and reasonable critiques of reports that are focused on these processes. These guidelines are not meant to be a dogmatic set of rules, because the appropriateness of any assay largely depends on the question being asked and the system being used. Moreover, no individual assay is perfect for every situation, calling for the use of multiple techniques to properly monitor autophagy in each experimental setting. Finally, several core components of the autophagy machinery have been implicated in distinct autophagic processes (canonical and noncanonical autophagy), implying that genetic approaches to block autophagy should rely on targeting two or more autophagy-related genes that ideally participate in distinct steps of the pathway. Along similar lines, because multiple proteins involved in autophagy also regulate other cellular pathways including apoptosis, not all of them can be used as a specific marker for bona fide autophagic responses. Here, we critically discuss current methods of assessing autophagy and the information they can, or cannot, provide. Our ultimate goal is to encourage intellectual and technical innovation in the field

Erratum to: Guidelines for the use and interpretation of assays for monitoring autophagy (3rd edition) (Autophagy, 12, 1, 1-222, 10.1080/15548627.2015.1100356

No abstract available