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In mechanical engineering, one-degree-of-freedom systems form the basis of a wide range of electrical and mechanical components. Using these models, engineers can predict the behavior of various parts in a larger system, which gives them insight into how different forces interact with each other.
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A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...
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Related Experiment Video

Updated: May 5, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Entropy and convexity for nonlinear partial differential equations.

John M Ball1, Gui-Qiang G Chen

  • 1Mathematical Institute, University of Oxford, , Oxford, OX2 6GG, UK.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|November 20, 2013
PubMed
Summary
This summary is machine-generated.

This article introduces entropy and convexity, fundamental mathematical concepts, and their application in analyzing nonlinear partial differential equations. These nonlinear methods are crucial for understanding complex systems in science and engineering.

Keywords:
convexitydiscontinuous solutionsentropynonlinear methodspartial differential equationssingular solutions

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Area of Science:

  • Mathematics
  • Thermodynamics
  • Statistical Physics

Background:

  • Partial differential equations (PDEs) are essential for modeling phenomena across physical, chemical, biological, and social sciences.
  • Entropy, originating in thermodynamics, and convexity, a mathematical concept for sets and functions, are foundational ideas.
  • Both concepts have evolved significantly, becoming central to modern mathematics.

Discussion:

  • Nonlinear methods employing entropy and convexity are increasingly vital for analyzing nonlinear PDEs.
  • This article serves as an introduction to these powerful analytical tools.
  • It also previews the scope and contributions of related research papers.

Key Insights:

  • Entropy and convexity provide essential frameworks for understanding and solving complex nonlinear PDEs.
  • The integration of these concepts offers advanced techniques for mathematical analysis.
  • Their application spans diverse scientific and engineering disciplines.

Outlook:

  • Continued development and application of entropy and convexity in nonlinear analysis.
  • Exploring new frontiers in PDEs through these established mathematical principles.
  • Fostering interdisciplinary research leveraging these analytical methods.