Search · four archives
Search · four archives
24 papers · ranked by Valyu relevance
Per Kristen Jakobsen
A partial differential equation(PDE), is an equation involving one or more functions of two or more variables, and their partial derivatives, up to some finite order. Here are some examples (4-1)$u_{x}+u_{y}\quad=0,\tag{1}$ $u_{x}+yu_{y}=0$, (4-2)$\left(u_{x}\right)^{2}+\left(u_{y}\right)^{2}=1,\tag{3}$ where…
Yuxin Wang, Yueyang Shen, Daxuan Deng, Ivo D. Dinov
Historically, PDEs have been classified as ‘‘elliptic’’, ‘‘parabolic’’, or ‘‘hyperbolic’’.9,11,12 Naturally, most of the fundamental PDEs, namely the Laplace equation, heat equation, and wave equation, fall into these types of categories. Contrasting the differences between separate partial differential equations may…
Hossein Jafari, Chaudry M. Khalique, Dumitru Baleanu
In recent years, the partial differential equations, both fractional and integer orders, have been recognized as a powerful modeling methodology. They are inspired by problems which arise in diverse fields such as biology, fluid dynamics, physics, differential geometry, control theory, materials science, and…
Steven L. Brunton, J. Nathan Kutz
Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This review will examine several promising avenues of PDE research that are being…
Xiaoping Xu
| | Preface | | v | | --- | --- | --- | --- | | | Introduction | | vii | | | | Notational Conventions xviii | | | I | | Ordinary Differential Equations | 1 | | 1 | | First-Order Ordinary Differential Equations | 3 | | | 1.1 | Basics | 3 | | | 1.2 | Special Equations | 9 | | 2 | | Higher-Order Ordinary Differential…
Noureddine Mhadhbi, Sameh Gana, Mazen Fawaz Alsaeedi
This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the form…
Jianfeng Wang
In this paper we discuss the first order partial differential equations resolved with any derivatives. At first, we transform the first order partial differential equation resolved with respect to a time derivative into a system of linear equations. Secondly, we convert it into a system of the first order linear…
K. V. Zhukovsky
We propose operational method with recourse to generalized forms of orthogonal polynomials for solution of a variety of differential equations of mathematical physics. Operational definitions of generalized families of orthogonal polynomials are used in this context. Integral transforms and the operational exponent…
Alexander Krikun
These are the notes for a series of Numerical Study group meetings, held in Lorentz institute in the fall of 2017. The aim of the notes is to provide a nonspecialist with the minimal knowledge in numerical methods used in BVP for PDEs, necessary to solve the problems typically arising in applications of holography to…
Timilehin Kingsley Akinfe, Adedapo Chris Loyinmi
In this research, an unrivalled hybrid scheme which involves the coupling of the new Elzaki integral transform (an improved version of Laplace transform) and a modified differential transform called the projected differential transform (PDTM) have been implemented to solve the generalized Burgers-Fisher's equation…
Elif Deniz, Adnan Rashid, Osman Hasan, Sofiène Tahar
Partial Differential Equations (PDEs) are widely used for modeling the physical phenomena and analyzing the dynamical behavior of many engineering and physical systems. The heat equation is one of the most well-known PDEs that captures the temperature distribution and diffusion of heat within a body. Due to the wider…
Jurgen Riedel, Chris P. Barnes
In this study we examine the emergence of complex biological patterns through the lens of reaction-diffusion systems. We introduce two novel complexity metrics, Diversity of Number of States (DNOS) and Diversity of Pattern Complexity (DPC), which aim to quantify structural intricacies in pattern formation, enhancing…
Jurgen Riedel, Chris P. Barnes
In this study we examine the emergence of complex biological patterns through the lens of reaction-diffusion systems. We introduce two novel complexity metrics — Diversity of Number of States (DNOS) and Diversity of Pattern Complexity (DPC)— which aim to quantify structural intricacies in pattern formation, enhancing…
Jonathan R. Potts, Ulrike E. Schlägel
Mathematical analysis of partial differential equations (PDEs) has led to many insights regarding the effect of organism movements on spatial population dynamics. However, their use has mainly been confined to the community of mathematical biologists, with less attention from statistical and empirical ecologists. We…
Stuart T. Johnston, Ruth E. Baker, Sean D.L McElwain, Matthew J. Simpson
Invasion processes are ubiquitous throughout cell biology and ecology. During invasion, individuals can become isolated from the bulk population and behave differently. We present a discrete, exclusion-based process that models the birth, death and movement of individuals. The model distinguishes between individuals…
Jasmine A. Nirody, Padmini Rangamani
Mathematical modeling is now used commonly in the analysis of signaling networks. With advances in high resolution microscopy, the spatial location of different signaling molecules and the spatio-temporal dynamics of signaling microdomains are now widely acknowledged as key features of biochemical signal transduction.…
Authors not listed
Redox-mediated flow batteries (RMFBs) promise increased energy density through the incorporation of solid active materials into the external tanks, but the operation and design of these devices is challenged by kinetic, thermodynamic, and transport complexities introduced by the solid-mediator reactions. Here, we…
Per Kristen Jakobsen
The amplitude equations (190) looks complicated, but they are special in the sense that they can be solved exactly. We have noted before that the amplitude equations that appears when we use the method of multiple scale tends to be special. We will see more of this later when we apply the method to partial differential…
Nikolay K. Vitanov, Quanmin Zhu, António Lopes
Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins…
Stephen Coleman
This research examines the geographical distributions of several historical epidemics in the United States and investigates whether they reached a geographical equilibrium, however briefly. An equilibrium distribution over a geographical area, as the end state of a diffusion or spatial contagion process, has definitive…
Authors not listed
Real-world datasets in chemical engineering and bioengineering processes--such as those from catalytic reactors, multiphase flows, polymerization reactors, bioreactors, and clinical trials--can often be unlabelled or disorganized, rendering the training of existing supervised learning models ineffective at learning the…
Authors not listed
The rapid depressurisation of pressure vessels containing hazardous substances in chemical plants, known as blowdown, is a critical process for ensuring plant safety. Blowdown significantly reduces the inventory as well as duration and rate of potential leaks, thereby mitigating the risks of escalation, fire and…
José Augusto Fontenele Magalhães, Muhammad Fuady Emzir, Francesco Corona
In order to characterise the dynamics of a biochemical system such as the chemostat, we consider a differential description of the evolution of its state under environmental fluctuations. We present solutions to the filtering problem for a chemostat subjected to geometric Brownian motion. Under this modelling…
Authors not listed
A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptomatic solution of the Schrodinger equation and an unknown function. By inserting this trial…