Multistability, oscillations and travelling waves in a product-feedback autocatalator model. II The initiation and propagation of travelling waves

1994 ◽  
Vol 349 (1691) ◽  
pp. 389-415 ◽  
Keyword(s):  
Initial Value Problem ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Travelling Waves ◽  
Temporal Stability ◽  
Kinetic Scheme ◽  
Travelling Wave ◽  
Initial Value ◽  
General Properties ◽  
Initiation And Propagation

The initiation and propagation of reaction diffusion travelling waves in a model product-feedback autocatalator kinetic scheme, for which the well-stirred (spatially homogeneous) case has been considered previously in detail, in discussed. A priori bounds and general properties are obtained for the initial-value problem. These are extended by numerical solutions of the initial-value problem in the three, previously identified expanding, contracting and reduced cases. In the reduced case, the concentrations attained at the rear of the wave are seen to depend on the existence and temporal stability of a non-trivial steady state of the system. In the contracting case, a constant non-zero value is achieved by the concentration of reactant A, while the concentrations of autocatalyst B and reactant C are pulse-like and tend to zero at long times. In the expanding case, a propagating wave front is seen, but now the concentrations behind the wave can grow in time. The permanent form travelling wave equations are then treated. General properties of their solutions are obtained and asymptotic solutions, valid for large initial concentrations of reactant A, are derived in all three cases.

The development of travelling waves in a simple isothermal chemical system - I. Quadratic autocatalysis with linear decay

1989 ◽  
Vol 424 (1866) ◽  
pp. 187-209 ◽  
Keyword(s):  
Initial Value Problem ◽  
Numerical Solutions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Chemical System ◽  
Length Scale ◽  
Diffusion Front ◽  
Initial Value ◽  
First Case

The possibility of travelling reaction–diffusion waves developing in the chemical system governed by the quadratic autocatalytic or branching reaction A + B → 2B (rate k 1 ab ) coupled with the decay or termination step B → C (rate k 2 b ) is examined. Two simple solutions are obtained first, namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the reactant B. Both of these indicate that the criterion for the existence of a travelling wave is that k 2 < k 1 a 0 , where a 0 is the initial concentration of reactant A. The equations governing the fully developed travelling waves are then discussed and it is shown that these possess a solution only if this criterion is satisfied, i. e. only if k = k 2 / k 1 a 0 < 1. Further properties of these waves are also established and, in particular, it is shown that the concentration of A increases monotonically from its fully reacted state at the rear of the wave to its unreacted state at the front, while the concentration of B has a single hump form. Numerical solutions of the full initial value problem are then obtained and these do confirm that travelling waves are possible only if k < 1 and suggest that, when this condition holds, these waves travel with the uniform speed v 0 = 2√ (1 – k ). This last result is established by a large time analysis of the full initial value problem that reveals that ahead of the reaction–diffusion front is a very weak diffusion-controlled region into which an exponentially small amount of B must diffuse before the reaction can be initiated. Finally, the behaviour of the travelling waves in the two asymptotic limits k → 0 and k → 1 are treated. In the first case the solution approaches that for the previously discussed k = 0 case on the length scale associated with the reaction–diffusion front, with the difference being seen on a much longer, O ( k –1 ), length scale. In the latter case we find that the concentration of A is 1 + O (1 – k ) and that of B is O ((1 – k ) 2 ), with the thickness of the reaction–diffusion front being of O ((1 – k ) ½ ).

The development of travelling waves in a simple isothermal chemical system II. Cubic autocatalysis with quadratic and linear decay

1990 ◽  
Vol 430 (1879) ◽  
pp. 315-345 ◽  
Keyword(s):  
Numerical Solutions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Chemical System ◽  
Necessary Condition ◽  
Travelling Wave Solution ◽  
Diffusion Front ◽  
Initial Value ◽  
Linear Decay

The possibility of travelling reaction-diffusion waves developing in the isothermal chemical system governed by the cubic autocatalytic reaction A + 2B → 3B (rate k 3 ab 2 ) coupled with either the linear decay step B → C (rate k 2 b ) or the quadratic decay step B + B → C (rate k 4 b 2 ) is examined. Two simple solutions are obtained,namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the autocatalyst B. Both of these suggest that, for the quadratic decay case, a wave will develop only if the non-dimensional parameter k ═ k 4 / k 3 a 0 < 1 (where a 0 is the initial concentration of the reactant A), with there being no restriction on the initial input of the autocatalyst B. However, for the linear decay case the initiation of a travelling wave depends on the parameter v ═ k 2 / k 3 a 2 0 and that, in addition, there is an input threshold on B before the formation of a wave will occur. The equations governing the fully developed travelling waves are then considered and it is shown that for the quadratic decay case the situation is similar to previous work in quadratic autocatalysis with linear decay, with a necessary condition for the existence of a travelling-wave solution being that K < 1. However, the case of linear decay is quite different, with a necessary condition for the existence of a travelling wave solution now found to be v < 1/4 Numerical solutions of the equations governing this case reveal further that a solution exists only for v < v c , with v c ≈ 0.0465, and that there are two branches of solution for 0 < v < v c . The behaviour of these lower branch solutions as v → 0 is discussed. The initial-value problem is then considered. For the quadratic decay case it is shown that the uniform state a ═ a 0 , b ═ 0 is globally asymptotically stable (i. e. a → a 0 , b → 0 uniformly for large times) for all k > 1. For the linear decay case it is shown that the development of a travelling wave requires β 0 > v (where β 0 is a measure of the initial input of B) for v < v c . These theoretical results are then complemented by numerical solutions of the initial-value problem for both cases, which confirm the various predictions of the theory. The behaviour of the solution of the equations governing the travelling waves is then discussed in the limits K → 0, v → 0 and K → 1. In the first case the solution approaches the solution for K ═ 0 (or v =0) on the length scale of the reaction-diffusion front, with there being a long tail region of length scale O ( K -1 ) (or O ( v -1 )) in which the autocatalyst B decays to zero. In the latter case we find that the concentration of reactant A is 1 + O [(1 - k )] and autocatalyst B is O[(1 - k 2 ] with the thickness of the reaction-diffusion front becoming large, of thickness O [(1- k ) -3/2 ].

scholarly journals Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Initial Value Problem ◽  
Numerical Solutions ◽  
Oblate Spheroid ◽  
Initial Value ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Geodesic Equations ◽  
Research Article ◽  
Fast Solution ◽  
Geodesic Problem

AbstractThe direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.

The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. II. An initial-value problem with an immobilized or nearly immobilized autocatalyst

1991 ◽  
Vol 336 (1644) ◽  
pp. 497-539 ◽  
Keyword(s):  
Travelling Waves ◽  
Chemical Species ◽  
Travelling Wave ◽  
Small Time ◽  
Initial Value ◽  
Slab Geometry ◽  
Diffusion Rates ◽  
Large Spike ◽  
Reaction Schemes ◽  
Cubic Autocatalysis

We study the isothermal autocatalytic reaction schemes, A + B -> 2B (quadratic autocatalysis), and A + 2B → 3B (cubic autocatalysis), where A is a reactant and B is an autocatalyst. We consider the situation when a quantity of B is introduced locally into a uniform expanse of A, in one-dimensional slab geometry. In addition, we allow the chemical species A and B to have unequal diffusion rates D A and D B respectively, and study the two closely related cases, ( D B / D A ) = 0 and 0 < ( D B / D A ) < 1. When (D b /D a ) = 0 a spike forms in the concentration of B, which grows indefinitely, and we can obtain both large and small time asymptotic solutions. For 0 < ( D B / D A ) < 1 there is a long induction period during which a large spike forms in the concentration of B, before a minimum speed travelling wave is generated. We can relate the results for case ( D B / D A ) = 0 to the solution when 0 < ( D B / D A ) < 1 to obtain detailed information about its behaviour.

On the Benney equation

2009 ◽  
Vol 139 (6) ◽  
pp. 1121-1144 ◽  
Author(s):  
Amin Esfahani
Keyword(s):  
Sobolev Spaces ◽  
Initial Value Problem ◽  
A Priori ◽  
Travelling Wave ◽  
Well Posedness ◽  
Travelling Wave Solutions ◽  
Initial Value ◽  
Wave Solutions ◽  
Benney Equation ◽  
Well Posed

We study the Benney equation and show that the associated initial-value problem is locally well-posed in Sobolev spaces Hs(ℝ2) for s > −2. Furthermore, we use a priori estimates to establish the global well-posedness for s ≥ 0. We also prove that these results are in some sense sharp. In addition, we obtain some exact travelling-wave solutions of the equation.

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