Closed-Form Solutions to the Transient/Steady-State Navier-Stokes Fluid Dynamic Equations

2006 ◽  
Author(s):  
P. A. Murad
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Fluid Dynamic ◽  
Dynamic Equations ◽  
Navier Stokes ◽  
Closed Form Solutions ◽  
Stokes Fluid

scholarly journals On closed-form solutions of some nonlinear partial differential equations

2000 ◽  
Vol 23 (2) ◽  
pp. 81-88 ◽  
Author(s):  
S. S. Okoya
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Steady State ◽  
Closed Form ◽  
Nonlinear Wave Equation ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Thermal Explosion Theory

This paper is devoted to closed-form solutions of the partial differential equation:θxx+θyy+δexp(θ)=0, which arises in the steady state thermal explosion theory. We find simple exact solutions of the formθ(x,y)=Φ(F(x)+G(y)), andθ(x,y)=Φ(f(x+y)+g(x-y)). Also, we study the corresponding nonlinear wave equation.

BANG-BANG, IMPULSE, AND SUSTAINABLE HARVESTING IN AGE-STRUCTURED POPULATIONS

2012 ◽  
Vol 20 (02) ◽  
pp. 133-153 ◽  
Author(s):  
N. HRITONENKO ◽  
YU. YATSENKO
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Structured Populations ◽  
Policy Implications ◽  
Optimal Harvesting ◽  
Steady State Analysis ◽  
Sustainable Harvesting ◽  
Closed Form Solutions ◽  
Age Structured ◽  
Mckendrick Model

An optimal harvesting problem is analyzed in the Lotka–McKendrick model of age-structured populations. Depending on the imposed constraints, this problem possesses bang-bang or impulse regimes, which have meaningful interpretations and relevant policy implications. A steady state analysis of the model produces closed-form solutions for the optimal sustainable harvesting in a new and an existing population.

scholarly journals Dynamic Catalysis Fundamentals: I. Fast calculation of limit cycles in dynamic catalysis

2021 ◽  
Author(s):  
Brandon Foley ◽  
Neil Razdan
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Closed Form ◽  
Linear Systems ◽  
Computational Cost ◽  
Kinetic Equations ◽  
Reaction Rates ◽  
Closed Form Solutions ◽  
Non Linear ◽  
Non Linear Systems

Dynamic catalysis—the forced oscillation of catalytic reaction coordinate potential energy surfaces (PES)—has recently emerged as a promising method for the acceleration of heterogeneously-catalyzed reactions. Theoretical study of enhancement of rates and supra-equilibrium product yield via dynamic catalysis has, to-date, been severely limited by onerous computational demands of forward integration of stiff, coupled ordinary differential equations (ODEs) that are necessary to quantitatively describe periodic cycling between PESs. We establish a new approach that reduces, by ≳108×, the computational cost of finding the time-averaged rate at dynamic steady state (i.e. the limit cycle for linear and nonlinear systems of kinetic equations). Our developments are motivated by and conceived from physical and mathematical insight drawn from examination of a simple, didactic case study for which closed-form solutions of rate enhancement are derived in explicit terms of periods of oscillation and elementary step rate constants. Generalization of such closed-form solutions to more complex catalytic systems is achieved by introducing a periodic boundary condition requiring the dynamic steady state solution to have the same periodicity as the kinetic oscillations and solving the corresponding differential equations by linear algebra or Newton-Raphson-based approaches. The methodology is well-suited to extension to non-linear systems for which we detail the potential for multiple solutions or solutions with different periodicities. For linear and non-linear systems alike, the acute decrement in computational expense enables rapid optimization of oscillation waveforms and, consequently, accelerates understanding of the key catalyst properties that enable maximization of reaction rates, conversions, and selectivities during dynamic catalysis.

Some Fundamental Thermoelastic Problems in Orthotropic Slabs

1966 ◽  
Vol 33 (1) ◽  
pp. 45-51 ◽  
Author(s):  
B. R. Baker
Keyword(s):  
Steady State ◽  
Temperature Distribution ◽  
Closed Form ◽  
Fourier Transforms ◽  
Broad Class ◽  
Complex Stress ◽  
Closed Form Solutions ◽  
Definite Form ◽  
Stress Components ◽  
Thermoelastic Problems

Steady-state problems of stresses in orthotropic slabs subjected to forces and temperature distribution on the faces are solved by means of Fourier transforms. The usual direct applications in the literature are shown for simple examples to lead to divergent integrals but, by introduction of generalized transforms, a very broad class of problems can be handled. As a part of these considerations, a more definite form of Saint-Venant’s principle is obtained. For a certain class of material constants, it is possible to obtain closed-form solutions for stresses when concentrated loads or temperature sources are applied to the faces of the slab. Results are presented for several examples in the form of complex stress potentials and graphs of the corresponding stress components.

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