This book will introduce methods for solving some problems of the forms (i) and (iii) in the context of continuous systems and differential equations. Do the solutions starting from other initial conditions follow the behavior of the solution from x0?).
The Motion of Particles
Problems that can be presented in the form (1.2) are called dynamical systems, where the independent variable t usually represents time, but can also represent other properties depending on the context of the system. This can give a complete characterization of the dynamics without having to attempt to explicitly construct the solutions.
Chemical Reaction Kinetics
This is called a zero-order reaction because the rate does not depend on the concentration of any reactants, k=k·1=k A0. This is called a first-order reaction, since the rate of the reaction depends linearly on the concentration of the individual reactant. ii) Transformation: A is consumed by B, which is produced from A A−→k B =⇒ d A. 1.10).
Ecological and Biological Models
The influence of prey consumption by predators is called predation and generates an additional loss term in the logistic equation (1.19) for the rabbit population. In the context of the spread of disease [18], simple models of epidemics divide the total population into subgroups, depending on whether individuals are infected (I(t)), susceptible to the disease (S(t)), or recovering from the disease. disease (R(t)).
One-Dimensional Phase-Line Dynamics
The characterization of the dynamics of the solutions nearx∗ based on the linearized equation (1.26) is commonly called linear stability analysis. This is consistent with the results of the local stability analysis at eachx∗ controlled by the sign f(x∗).
Two-Dimensional Phase Plane Analysis
Nullclines
Nullclines are not solutions of the system, but they provide valuable insight into the properties of solution trajectories that pass through nullclines (or lie to one side or the other). The intersection of the nullclines makes immediately clear the locations of the two equilibrium points.
Further Directions
The nature of these equilibria could be deduced from information provided by the nullclines without these slightly cumbersome algebraic calculations, see Fig.1.4. More detailed models of populations, called structured population models, divide the population by age or size; if treated as continuous variables, then the rate equations yield partial differential equations.
Exercises
We consequently formulate continuum models as partial differential equations (PDE) that govern the evolution of density functions (f(x,t)) that describe properties of the population at a given position and time. In each of these contexts, PDEs can be used to describe the redistribution of the property of interest over time.
The Reynolds Transport Theorem
By conveying shifts or "motion" in the density, either with respect to spatial position or with the independent variable x representing other properties (eg, velocity, age, size), such PDEs are also broadly called transport equations. The next stage in the formulation of a continuum model is to determine the rate of change of a property evaluated over a "lump of material"—in other words, we consider a specific set of particles, defined as starting from a set Awaardes (2.1a) occupying a regionDin space.
Deriving Conservation Laws
Then applying du Bois-Reymond's lemma yields the general conservation law, including reaction terms (also called source or sink terms, depending on whether the rate of production is positive or negative). 2.16) Equation (2.14) can be applied to Newton's second law, which states that the rate of change of momentum is equal to the sum of the applied forces. Applying Reynolds' transport theorem, Du Bois Reymond's lemma, and expressing the forces in terms of the divergence of a stress tensor, f = ∇ σ, yields the Cauchy momentum equation.
The Linear Advection Equation
Descriptions of wave properties in terms of the dispersion relation are essential in many models that emerge as generalizations of the advection equation. We now turn our attention to describing methods for obtaining solutions of various generalizations of the advection equation (2.19).
Systems of Linear Advection Equations
Solving this system yields the eigenvalues (λ) that define the wave velocities and the components of the eigenvectors that define the relation of w to the solutions and q, (2.25). Assuming test solutions of the form w=u·pwhereu=(A,B,C, . .)Ten the relationship between (2.26) and the coefficients in (2.23), we see that the general eigenvalue problem can be stated as.
The Method of Characteristics
As an example, consider the transportation problem forp(x,t). 2.35) After the above analysis, the corresponding characteristic problem is given by. Solving (2.36a) gives X(t;A)=Ae2t, which we can then substitute into the ODE for P(2.36b) to obtain the general solution.
Shocks in Quasilinear Equations
We observe that the part of the solution starting from x∈ [−1,0] spreads over an increasingly large region as its characteristics separate from each other (this is sometimes called a fanorrefraction expansion wave). Conversely, the part of the solution starting from x∈ [0,1] is being compressed into a smaller region (up to =1) and is called a compression wave.
Further Directions
Exercises
Show that the corresponding potential must be of the form φ(x,y,t)=B(y)sin(kx−ωt) and obtain the dispersion relation ω(k). Obtain a system of five autonomous ODEs for X, T, P, R, Q in terms of those variables and derivatives of F. c) Determine the characteristic equations for the general Hamilton-Jacobi PDE, pt+H(p,px,x ,t)=0,.
Review and Generalisation from Calculus
Functionals
For the most part, we will only consider functions that are definite integrals of a function and its derivatives. We will describe a straightforward procedure (as an extension of the process of finding critical points of an objective function) for determining optimal solutions of a function.
General Approach and Basic Examples
The Simple Shortest Curve Problem
It is important to note that the form of the first variation in (3.8) has been chosen to deliberately suggest that the perturbation function h(x) should appear linear and undifferentiated. The boundary conditions given by (3.11) eliminate the contributions of the boundary terms and leave the final form of the first variation of the functional as a single integral.
The Classic Euler – Lagrange Problem
Since this applies to all admissible choices of h(x), we obtain by (3.14) the differential equation ona≤x≤b,. Subject to the boundary conditions (3.18b), (3.21) defines a differential equation problem that must have a unique solution viry=y∗(x).
The Variational Formation of Classical Mechanics
Motion with Multiple Degrees of Freedom
A generalization of the Euler–Lagrange equation of motion is the derivation of the equations of motion for a multivariable system. The action can be written from the difference between the kinetic and potential energies.
The Influence of Boundary Conditions
Problems with a Free Boundary
Consequently, the Euler–Lagrange equation is again given by (3.21) and (in this case) leads to the same ODE as found for the prescribed endpoint case (3.16). However, the new boundary conditions for the Euler–Lagrange problem, y∗(0)=0 eny∗(1)=0, now choose a different solution, y∗(x)≡0 (corresponding to the line from the origin along the x-axis to the point of intersection with the line x=1).
Problems with a Variable Endpoint
62 3 Variational Principles Applying Leibniz's rule to (3.27), where ε plays the role of z, we obtain the first variation of the form. By expanding (3.29) in the limit ε→0, by matching the first terms in the corresponding expansions of the left and right sides, we obtain*(b*)=f(b*) for the optimal solution.
Optimisation with Constraints
Review of Lagrange Multipliers
If the parameter t is taken to represent time, the geometric interpretation of this equation is that the "velocity" vector dx/dt (which is tangent to the curve(t)) is perpendicular to the plane curve=0 since the gradient∇ yeast orthogonal to contours of constant function value. The above discussion satisfactorily motivates the introduction of the augmented (constrained) Lagrange objective function This is just a convenient form that renders the set of equations above as the critical point equations for L with respect to its three variables,.
Integral Constraints: Isoperimetric Problems
66 3 Variational Principles Once this functional has been identified, we follow the standard variational process described in Section 3.2, starting with introducing perturbations to all unknowns. This again shows that the form of the Euler-Lagrange equation (3.21) generalizes to a wide set of problems.
Geometric Constraints: Holonomic Problems
To complete the solution of the constrained problem, the arc length constraint, s(λ∗)=S, must be applied to determine a value for λ∗ in terms of S. It can be shown that the appropriate generalization of (3.34) to holonomic problems is the augmented Lagrangian. 3.42) Here a notable difference from (3.38) for isoperimetric problems is that the Lagrange multiplier is a function oft rather than a constant, and cannot be factored from the integral as we did in (3.37).
Differential Equation Constraints: Optimal Control
Similarly, the first equation gives an ordinary differential equation for the Lagrangian evolution. Starting from the algebraic relation, we can eliminate the control in terms of co-state,u= −λ.
Further Directions
To illustrate the dramatic influence of control, the optimally controlled solution grows exponentially, while the uncontrolled solution degrades exponentially. Although the introductory presentation in this chapter focused on formulating problems leading to ODEs, some exercises will show that PDEs can be derived from multiple integrals in the same way.
Exercises
Show that the Euler–Lagrange equations for x,y give the parametric equations of the line found as a solution to (3.10a,3.10b). The length of the beam is (domain: 0≤ x ≤ ) with constant mass densityρ (mass per unit length), constant bending stiffnessE (as the spring constant) and constant moment of inertiaI.
Dimensional Quantities
The SI System of Base Units
These dimensional units are fundamental in the sense that they describe independent physical properties that cannot be represented in terms of each other. Dimensional units that can be expressed in terms of basic units are known as derived units.
Dimensional Homogeneity
The Process of Nondimensionalisation
Projectile Motion
The solution of (4.7a) will be a function of the independent variable and the remaining non-dimensional constantsΠ2andΠ4, namely. We have effectively reduced the complexity of the original problem (involving a solution function of five variables) to a solution that depends on only three variables.
Terminal Velocity of a Falling Sphere in a Fluid
The solution to the original dimensional problem can then be written in terms of (4.15) and (4.12) as As already described in connection with (4.8), the solution to this is to change the size of the problem.
The Burgers Equation
In (4.22) we have three scaling constants whose values can be set (any choice) of three of the six Π in (4.23) to unity, say. Conversely, without using scaling analysis, the best that can be said in general about the solution to the original problem (4.21) is that it will have some dependence on each parameter and variable.
Further Applications of Dimensional Analysis
Projectile Motion (Revisited)
The introduction of the drag constant of proportionality K, also provides another way to generate a quantity with dimensions of time and this allows us to write. So (4.34) gives us insight into the qualitative dependence of T on all quantities in the system and a more detailed quantitative solution would involve the numerical solution of (4.36).
Closed Curves in the Plane
The Buckingham Pi Theorem
Mathematical Consequences
Writing the solution to the original problem in terms of this dimensionless solution requires characteristic scales for all dependent and independent variables. Similar to (4.42), the choices for these scales given the problem quantities are obtained by solving for the exponents v.
Application to the Quadratic Equation
Since [Π1] =1, we see that [T] = [B/A], namely that the ratio between the given quantities B/A determines a characteristic scale. Although the Buckingham theorem cannot predict the details of the function f(Π2), (4.47) has indicated the way in which the solution of (4.45) depends on the given quantities.
Further Directions
Standard theory from linear algebra then gives that each solution will have two free parameters that can be set arbitrarily, with two variables being defined in terms of the other two variables. Finally, in our introductory discussion we have barely touched on the relation of dimensional analysis to similarity and scale models—how to build a model ship, for example, with properties that when scaled up will match those of the actual ship.
Exercises
Determine the choices for L,T that give an ODE for xw by normalizing the three terms on the left side of the equation. This problem can be described in terms of the set of Parameters:. called Reynolds, Bond and Capillary numbers) or in group terms.
Solution Techniques
Finding Scaling-Invariant Symmetries
Setting UT/L =1 and UL =1 eliminates the scaling constants of (5.4a,5.4b) and makes the scaling system identical to (5.3a,5.3b) and thus the problem is scale invariant. There is a strong similarity between the scale forms (5.1) and (4.22), but there is also an important underlying difference.
Determining the Form of the Similarity Solution
The similarity function is a scale-invariantΠ that is (more simply) linearly related to the model solution, that is, =1 in (5.7). We note that using this representation of the solution reduces the boundary condition atx =0 strongly >0 to the condition f(0)=0.
Solving for the Similarity Function
The boundary condition atx=0 would then be indeterminate (and thus more difficult to work with). Finally, we note that satisfying the integral condition in (5.12) necessitates the use of both solutions from (5.13) to construct the solution in (5.3) (compare with Sect.2.6).
Further Comments on Self-Similar Solutions
118 5 Self-similar scaling solutions of differential equations If we insert a non-trivial solution, f(η) = η, into (5.11), we get a self-similar PDE solution of the form
Similarity Solutions of the Heat Equation
- Source-Type Similarity Solutions
- The Boltzmann Similarity Solution
In the context of the linear diffusion equation, we will now consider how the choice of side conditions can affect the overall structure of the similarity solution. Note that (5.15) is also subtranslation invariant in space,x = ˜x+a, and time, t = ˜t+b; namelyu˜=u(x,˜ t)˜ is also a solution of the heat equation for any choice of displacement and b.
A Nonlinear Diffusion Equation
5.2 (Left) Theu(x,t)reduction/spread of self-similar solution profiles (5.23) at various times, (Right) The profiles scaled in terms oft1/5u= fandt−1/5x=η showing the similarity function. The general solution of the differential equation can be obtained in closed form and, upon application of the boundary condition atη=0, reduces to.
Further Directions
Exercises
Find α, β, γ for the similar solution to this problem and note the ODEs that satisfy f(η)eng(η). Find the similarity solution to this problem and use it to determine the derivative of the solution at the boundary,ux(0,t) )=ctd .
Asymptotic Analysis: Concepts and Notation
Perturbation methods provide a systematic approach to constructing approximate solutions to equations such as. 6.1) in the limit of a vanishing small disturbance parameter, ε→0. There is also a "little oh" relation, "f =o(g)" asε→0 (also written as "f g") that describes f being asymptotically smaller than poison.
Asymptotic Expansions
- Divergence of Asymptotic Expansions
We know from calculus that in the limit ε → 0 it is possible to obtain an accurate estimate of the value of x(t) in the neighborhood of t∗ from a limited number of terms in the expansion. In summary, asymptotic expansions are always accurate in the limit ε→0, but for finite ε there will be a suboptimal truncation n=0,1,.
The Calculation of Asymptotic Expansions
- The Expansion Method
- The Iteration Method
- Further Examples
Our next example introduces a problem for which the expansion method using the asymptotic expansion (6.17) fails, and consequently illustrates the strengths of the iteration method. This indicates that our choice for the expansion of x was wrong, namely that the solutions of (6.23) are not of the assumed form (6.17).
A Regular Expansion for a Solution of an ODE Problem
Note that the asymptotic order of the expansion breaks down with O(x0)=O(εx1)=O(1/ε) when =O(1/√ .ε) and the construction of the solution would also break down at this point , since the assumption that|εx1| |x0|implicit in (6.25) would be violated. Such a split indicates a transition in scaling regimes, and solving the problem involves identifying the appropriate new scaling.
Singular Perturbation Problems
- Rescaling to Obtain Singular Solutions
Which solutions are obtained depends on the scale of the problem and the form of the asymptotic expansion assumed for the solution. This leading order equation thus yields only one of the two roots expected from the quadratic equation (6.27).
Further Directions
A particular difference between the iteration method described in Section 6.3.2 and the current methodology is worth noting. The iteration method attempts to identify one consecutive term in the asymptotic expansion at each iteration.
Exercises
While it can be very difficult to directly solve the full problem on an entire domain, constructing "partial solutions" for different regions using perturbation extensions can be straightforward. Further analysis is then necessary to combine the partial solutions into a complete solution of the entire problem.
Observing Boundary Layer Structure in Solutions
This limiting form of the solution satisfies the boundary condition atx =1, but not that atx=0 (7.1b). For most of the domain (here, kux = O(1)), the solution is given by (7.4) and is called the outer solution.
Asymptotics of the Outer and Inner Solutions
Since the original full problem (7.1a,7.1b) has a unique solution; if the inner and outer solutions are both valid in the overlap domain, they cannot be two different solutions and must in fact be two different asymptotic representations of the same solution. At the formal level, we have y∼ y0 over most of the domain, with the inner solution becoming important only in the inner domain.
Constructing Boundary Layer Solutions
- The Outer Solution
- The Distinguished Limits
- The Inner Solution
- Asymptotic Matching
- The Composite Solution
The location of the boundary layer and which boundary conditions apply to the interior solution may not be determined before fitting is applied. Consequently, the boundary layer must take the form given by (c) where the neglected term (3) is subdominant to the forward balance with O(1)O(ε−1).
Further Examples
Since the limit of the outer solution is O(1), so must the limit of the inner solution, i.e. β =0. Obtaining an exact solution to higher orders would involve obtaining further conditions on the expansions of the inner and outer solutions.
Further Directions
Matched asymptotics provide insight into whether the boundary layers are necessary to determine properties of the outer solution or simply correct the solution in narrow regions. Often, knowledge of the physical system being modeled can guide expectations of boundary layer positions and dominant equilibria; this can sometimes simplify the mathematical steps.
Exercises
For each case determine the α, β for each respective limit and write its corresponding leading order equation for Y0(X). The leading order outer solution isy0(x)≡4. a) Determine the leading order inner solution for the boundary layer atx∗=0.
The Classic Separation of Variables Solution
In contrast, the Neumann boundary conditions give the value of the directional derivative of the solution normal to the boundary. This is an elementary "building block" problem where the shape of the solution will be entirely due to the single inhomogeneous boundary condition.
The Dirichlet Problem on a Slender Rectangle
The boundary condition at X =0 is unchanged from (8.12c), but now the remaining boundary condition is provided by an analogue of the asymptotic fit to the outer solution (see d). This corresponds to the form of the Dirichlet "building block" problem with a single inhomogeneous boundary condition described in Section 8.1.
The Insulated Wire
This determines the boundary condition on the outer solution (8.25) in terms of the average of theg0. Analogous analysis of the boundary layer atx∗ =1 gives the boundary condition forC2(1) in terms of the mean value of g1 and determines the outer solution as.
The Nonuniform Insulated Wire
Finally, taking the dot product with the gradient ∇U yields the condition. depending on the preconditions. 8.33d). Boundary conditions for (8.37) are then determined by solving for the boundary layers atx∗L=0 and xR∗ =1 using separation of variables as was done in Section 8.3 to construct the uniform solution for the whole domain (8.20).
Further Directions
Exercises
Investigate the expansion of the boundary layers to show that if we want to find a solution uniformly valid for O(ε2), then the expansionu∼u0+εu1+ε2u2. If we understand the behavior of the system for a single cycle of the oscillation, we can determine how the perturbative forcing terms affect the problem cumulatively over long times and many oscillation periods.
Review of Solutions of the Linear Problem
186 9 Weakly Nonlinear Oscillators Two perturbation methods will be described to illustrate how weak effects can be incorporated into the leading-order solution to obtain more accurate long-term predictions of oscillatory behavior. We will see that the appearance of such terms is a central issue to be addressed in constructing accurate long-term asymptotic solutions for perturbed oscillators.
The Failure of Direct Regular Expansions
The respective periodic and decaying oscillatory behavior of these two solutions is shown in Fig.9.2 together with plots of the expansions (9.8) and (9.9). Applying the fundamental assumption of asymptotic ordering to the terms in (9.7), the expansions are valid only when.
Poincare – Lindstedt Expansions
While the example above was a linear problem, the Poincare-Lindstedt method extends directly to nonlinear equations with perturbation terms of the formεf(x). In fact, this is only possible when x(t) is a periodic solution, leading to a degenerate linear system for the coefficients of the resonance force expressions.
The Method of Multiple Time-Scales
Noting that the general solution of the leading-order problem has two independent solution terms, x˜0 = Asinθ +Bcosθ, and that each leads to resonant force in (9.19), it should be little surprising that the coefficients of both resonant terms can be set to zero with only one degree of freedom, ω1. Therefore, the Poincare–Lindstedt method can only be used to obtain periodic solutions and cannot, for example, generate the slowly decaying solution (9.11).
Further Directions
Exercises
Strongly-Nonlinear Oscillators: The van der Pol Equation
Complex Chemical Reactions: The Michaelis-Menten
Further Directions
Exercises
The Method of Moments
Turing Instability and Pattern Formation
Taylor Dispersion and Enhanced Diffusion
Further Directions
Exercises
Case Studies
Lubrication Theory
Dynamics of an Air Bearing Slider
Rivulets in a Wedge Geometry
- Imbibition in a Vertical Wedge
- Draining in a Vertical Wedge
