Matched Asymptotic Expansions in Reaction-Diffusion Theory

by Leach, James A.; Needham, D. J.

• ISBN13: 9781852337674
• ISBN10: 1852337672
• Format: Hardcover
• Copyright: 2004-01-01
• Publisher: Springer Verlag

Summary

This volume contains a wealth of results and methodologies applicable to a wide range of problems arising in reaction-diffusion theory. The first part is concerned with obtaining the complete structure of the large-time solution of scalar reaction diffusion equations, and systems of reaction-diffusion equations. The second part is concerned with the analysis of a class of singular reaction-diffusion equations.In this detailed analysis, use is made of the method of matched asymptotic expansions, dynamical systems theory, and comparison theorems, which provide a powerful combination of techniques for the detailed analysis of this broad class of reaction-diffusion equations.The monograph can be viewed both as a handbook, and as a detailed description of the methodology. Researchers in reaction-diffusion theory, and scientists applying reaction-diffusion theory to such areas as chemical kinetics, biological systems, epidemiology and population dynamics, will find it a popular addition to the literature.

Table of Contents

Part I The Evolution of Travelling Waves in Scalar Fisher-Kolmogorov Equations
Introduction	3	(12)
Generalized Fisher Nonlinearity	5	(5)
mth-Order (m > 1) Fisher Nonlinearity	10	(5)
Generalized Fisher Nonlinearity	15	(24)
Permanent Form Travelling Waves	15	(2)
Asymptotic Solution to IBVP as t → ∞	17	(18)
Case (III): v* > 2	18	(13)
Case (II): v* = 2, uT(Z,v*) ~ B* e-z as z → ∞	31	(2)
Case (I): v* = 2, uT(Z,v*) ~ A* e-z as z → ∞	33	(1)
Conclusions	34	(1)
Example: Fisher's General Genetic Population Model	35	(4)
mth-Order (m > 1) Fisher Nonlinearity: Initial Data with Exponential Decay Rates or Compact Support	39	(36)
Permanent Form Travelling Waves	40	(1)
Evolution of Travelling Waves in [P, m]	41	(25)
Asymptotic Solution as t → 0 for 0 ≤ x < ∞	41	(2)
Asymptotic Solution as x → ∞ for t = O(1)	43	(1)
Asymptotic Solution as t → ∞ in the Case of Initial Data with Exponential Decay Rate as x → ∞	43	(17)
Asymptotic Solution as t → ∞ in the Case of Initial Data with Compact Support	60	(6)
Numerical Solutions	66	(3)
Numerical Solutions when the Initial Data has Exponential Decay Rate as x → ∞	67	(2)
Numerical Solutions when the Initial Data has Compact Support	69	(1)
Summary	69	(3)
Initial Data with Exponential Decay as x → ∞	69	(2)
Initial Data with Compact Support	71	(1)
Consideration of a More General Class of Initial Data with Exponential Decay Rate as → ∞	72	(3)
mth-Order (m > 1) Fisher Nonlinearity: Initial Data with Algebraic Decay Rates	75	(36)
Permanent Form Travelling Waves	75	(1)
Asymptotic Solution as t → 0 for 0 ≤ x < ∞	76	(1)
Asymptotic Solution as x → ∞ for t = O(1)	77	(2)
α = 1/m-1	77	(1)
α > 1/m-1	78	(1)
Asymptotic Solution as t → ∞ when α = 1/m-1	79	(17)
1 < m < 2	79	(9)
m = 2	88	(3)
m > 2	91	(5)
Asymptotic Solution as t → ∞ when α > 1/m-1	96	(3)
Numerical Solutions	99	(7)
Summary	106	(5)
Extension to Systems of Fisher-Kolmogorov Equations. Example: A Simple Model for an Ionic Autocatalytic System	111	(40)
General Properties of Travelling Wave Solutions	114	(5)
Summary	119	(1)
The Existence of Travelling Wave Solutions	119	(3)
Equivalent Dynamical System	120	(2)
The Initial Value Problem (IVP)	122	(2)
Asymptotic Solution to IVP as t → ∞	124	(20)
Asymptotic Solution as t → 0	125	(4)
Asymptotic Solution as |x| → ∞	129	(1)
Asymptotic Solution as t → ∞	130	(12)
Summary	142	(2)
Numerical Solutions	144	(2)
Conclusions	146	(5)
Part II The Analysis of a Class of Singular Scalar Reaction-Diffusion Equations
Introduction	151	(4)
Permanent Form Travelling Waves (PTWs)	155	(22)
General Properties of PTW Solutions	155	(2)
PTW Solutions when m > n	157	(11)
Local Behaviour	159	(5)
Existence of a PTW Connection	164	(3)
Properties of the PTW Connection	167	(1)
Nonexistence of PTW Solutions when n > m	168	(3)
Nonexistence of PTW Solutions when n = m	171	(1)
Asymptotic Forms for c* (k)	171	(3)
k → 0+	171	(2)
k → kc-	173	(1)
Conclusions and Further Discussion	174	(3)
Extension of the Results on Nonexistence of PTW Solutions to a Wider Class of Singular Reaction-Diffusion Problems	175	(2)
The Initial-Boundary Value Problem	177	(36)
Existence, Uniqueness, and the Comparison Theorem	178	(1)
Comparison Theorem	178	(1)
Existence and Uniqueness for IBVP	178	(1)
Qualitative Behaviour of the Solution to IBVP' for m < n	179	(1)
Qualitative Behaviour of the Solution to IBVP for m = n	179	(3)
k > 1	179	(2)
k = 1	181	(1)
k < 1	182	(1)
Qualitative Behaviour of the Solution to IBVP for m > n	182	(9)
k > k*	183	(1)
kc < k ≤ k*	184	(3)
k = kc	187	(1)
0 < k < kc	188	(3)
Asymptotic Solution as t → 0 for 0 ≤ x < ∞	191	(10)
m < n	191	(1)
m = n	191	(10)
m > n	201	(1)
Summary	201	(1)
Asymptotic Solution as t → ∞ or as t → tc-	201	(8)
m < n	201	(1)
m = n	202	(4)
m > n	206	(3)
Conclusions	209	(4)
Asymptotic Solution of IBVP as t → 0 for 0 ≤ x < ∞: Initial Data with Exponential or Algebraic Decay Rates	213	(8)
Initial Data with Algebraic Decay as x → ∞	213	(3)
Initial Data with Exponential Decay as x → ∞	216	(3)
Summary and Further Discussion	219	(2)
Extension to the System of Singular Reaction-Diffusion Equations	221	(50)
The Well-stirred Case	223	(7)
The Phase Portrait	223	(3)
IVP1	226	(2)
IVP2	228	(1)
Summary	229	(1)
Asymptotic Solution as t → 0	230	(38)
m < n	230	(11)
m = n	241	(26)
m > n	267	(1)
Conclusions	268	(3)
A Construction of a Global Nonnegative Solution to the Scalar Equation wt = wxx + μ*wn	271	(2)
B Asymptotic Solutions to the Eigenvalue Problem (8.68)-(8.71) as m → 0+ and m → 1-	273	(4)
C Analysis of Boundary Value Problem (8.76)-(8.78)	277	(4)
D Analysis of Boundary Value Problem (8.90)-(8.92)	281	(2)
References	283	(6)
Index	289