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Related Concept Videos

Separable Differential Equations01:20

Separable Differential Equations

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...
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Reaction Mechanisms: The Steady-State Approximation01:26

Reaction Mechanisms: The Steady-State Approximation

The steady-state approximation, also referred to as the quasi-steady-state approximation to differentiate it from a true steady state, is a widely used method for simplifying calculations in complex reaction mechanisms. This approach is particularly useful when dealing with multi-step reactions that involve reverse reactions or several steps, which can significantly increase mathematical complexity and make the reactions nearly unsolvable analytically.The steady-state approximation operates on...
Linear Differential Equations01:27

Linear Differential Equations

The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law yields a...
Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...
Homogeneous Equilibria for Gaseous Reactions02:15

Homogeneous Equilibria for Gaseous Reactions

Homogeneous Equilibria for Gaseous Reactions
For gas-phase reactions, the equilibrium constant may be expressed in terms of either the molar concentrations (Kc) or partial pressures (Kp) of the reactants and products. A relation between these two K values may be simply derived from the ideal gas equation and the definition of molarity. According to the ideal gas equation:

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Planar Gradient Diffusion System to Investigate Chemotaxis in a 3D Collagen Matrix
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Global existence for semilinear reaction-diffusion systems on evolving domains.

Chandrasekhar Venkataraman1, Omar Lakkis, Anotida Madzvamuse

  • 1Department of Mathematics, University of Sussex, Mantell Building, Brighton, BN1 9RF, UK. c.venkataraman@sussex.ac.uk

Journal of Mathematical Biology
|February 5, 2011
PubMed
Summary

This study extends reaction-diffusion system existence results to evolving domains, offering broad applicability in pattern formation theory without growth rate sign constraints.

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Area of Science:

  • Mathematical Biology
  • Partial Differential Equations
  • Dynamical Systems

Background:

  • Reaction-diffusion systems are fundamental to modeling pattern formation.
  • Analysis on fixed domains is well-established.
  • Extending these models to dynamic environments presents significant challenges.

Purpose of the Study:

  • To establish global existence results for reaction-diffusion systems on evolving domains.
  • To generalize existing theory from fixed to dynamic spatial domains.
  • To provide a framework applicable to various pattern formation models.

Main Methods:

  • Development of analytical techniques for reaction-diffusion systems on time-dependent domains.
  • Extension of fixed-domain existence theorems to evolving spatial configurations.
  • Mathematical analysis without assumptions on the sign of growth rates.

Main Results:

  • Global existence of solutions is proven for reaction-diffusion systems on spatially linear, isotropically evolving domains.
  • The theoretical framework accommodates systems with diverse growth rate characteristics.
  • Numerical simulations confirm the validity of the derived existence results.

Conclusions:

  • The findings provide a robust theoretical foundation for studying pattern formation on dynamic substrates.
  • This work broadens the scope of applicability for reaction-diffusion models in biological and physical sciences.
  • The established results are crucial for understanding complex spatiotemporal dynamics in evolving systems.