An Introduction to Nonlinear Partial Differential Equations by J. David Logan

By J. David Logan

Compliment for the 1st variation:

"This ebook is definitely conceived and good written. the writer has succeeded in generating a textual content on nonlinear PDEs that isn't purely particularly readable but additionally obtainable to scholars from assorted backgrounds."
—SIAM Review

A useful creation to nonlinear PDEs and their real-world applications

Now in a moment variation, this well known ebook on nonlinear partial differential equations (PDEs) comprises extended assurance at the primary themes of utilized arithmetic in an basic, hugely readable structure and is offered to scholars and researchers within the box of natural and utilized arithmetic. This publication presents a brand new specialise in the expanding use of mathematical purposes within the lifestyles sciences, whereas additionally addressing key subject matters reminiscent of linear PDEs, first-order nonlinear PDEs, classical and susceptible recommendations, shocks, hyperbolic structures, nonlinear diffusion, and elliptic equations. not like related books that sometimes in basic terms use formal proofs and conception to illustrate effects, An advent to Nonlinear Partial Differential Equations, moment version takes a more effective method of nonlinear PDEs through emphasizing how the consequences are used, why they're vital, and the way they're utilized to actual problems.

The intertwining courting among arithmetic and actual phenomena is chanced on utilizing special examples of purposes throughout quite a few parts corresponding to biology, combustion, site visitors movement, warmth move, fluid mechanics, quantum mechanics, and the chemical reactor thought. New positive aspects of the second one version additionally include:

  • Additional intermediate-level workouts that facilitate the improvement of complicated problem-solving skills

  • New functions within the organic sciences, together with age-structure, trend formation, and the propagation of diseases

  • An increased bibliography that enables extra research into really good topics

With person, self-contained chapters and a large scope of insurance that gives teachers the pliability to layout classes to satisfy particular pursuits, An advent to Nonlinear Partial Differential Equations, moment version is a perfect textual content for utilized arithmetic classes on the upper-undergraduate and graduate degrees. It additionally serves as a useful source for researchers and pros within the fields of arithmetic, biology, engineering, and physics who wish to extra their wisdom of PDEs.

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Extra resources for An Introduction to Nonlinear Partial Differential Equations

Sample text

3. 3) by the Cole-Hopf transformatton u 4. Show that the PDE ut = --. 2Dv, v + kuu, + q(t)u = 0 can be reduced t o the inviscid Burgers' equation us using the transformation + uz', =0 1. Introduction t o Partial Differential Equations 34 Show that the same transformation transforms + kuu, + q(t)u ut - into cs where g = + tw, Du,, = O - g-l(s)uxx= 0. (k/D)exp ( - J q ( t ) d t ) . 5 . By rescaling, show that the porous media equation can be written in the form uT = (m> 2, (um)

The density and the flow rate, and therefore we need a constitutive equation that relates the two. t) is the pressure, and the positive constants g, p , and u are the acceleration due to gravity, the permeability. and the viscosity of the fluid, respectively. 18) is Darcy's law. and it is a basic assumption in many groundwater problems. It is physically plausible because it confirms our intuition that the flow rate should depend on the pressure gradient as well as gravity. Darcy's law is a statement replacing a momentum balance law.

Then dx g’(t) = u x x z + U,t = (ut + uu,), - u: = -u, 2 = -g(t) 2 The general solution of g’ = -g2 is g(t) = But g ( 0 ) = 1/c = -2 0. 0 + The nonexistence of a global solution to the initial value problem is a typically nonlinear phenomenon.

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