MTH2032 Chap.12 Characteristics for Second-Order Semilinear PDEs
Characteristics for Second-Order Semilinear PDEs
Week 12 returns to characteristics for second-order semilinear PDEs. Factorising the second-order operator leads to the quadratic Aλ² − Bλ + C = 0, called the auxiliary equation, whose roots give the characteristic slopes and repeat the hyperbolic, parabolic and elliptic classification.
Changing to characteristic coordinates produces canonical forms that can often be integrated directly, and the wave equation yields d'Alembert's solution as two shapes travelling in opposite directions. Each answer is checked by substituting the general solution back into the equation.
What this chapter covers
- 01
Factorising the second-order operator
- 02
Auxiliary equation and its sign
- 03
Hyperbolic canonical form u_vw
- 04
Parabolic canonical form u_ww
- 05
Elliptic equations and Poisson form
- 06
D'Alembert's solution
Worked example · free
Characteristics of a wave equation with speed two
- 1A = 1, B = 0, C = −4, so B² − 4AC = 16, which is positive, so the equation is hyperbolic.
- 1The auxiliary equation λ² − 4 = 0 has roots λ = 2 and λ = −2.
- 1Set v = y − 2x and w = y + 2x; the canonical form is u_vw = 0.
- 1Integrating gives u = F(y − 2x) + G(y + 2x).
Key terms
- Auxiliary Equation
- The quadratic Aλ² − Bλ + C = 0 whose roots give the characteristic slopes of a second-order PDE.
- D'Alembert Solution
- The general solution u = φ(x + ct) + ψ(x − ct) of the one-dimensional wave equation.
- Domain of Dependence
- The interval of initial data that can influence the solution of the wave equation at a given point.
- Characteristic Coordinates
- New variables v and w that stay constant along the two families of characteristics of a second-order PDE.
Characteristics for Second-Order Semilinear PDEs FAQ
Why is there a minus sign in the auxiliary equation?
Expanding the factorised operator gives λ1 + λ2 = B/A and λ1λ2 = C/A, so the roots solve Aλ² − Bλ + C = 0. The notes warn that the minus sign is the part most often forgotten. With the wrong sign the characteristic slopes are reversed and the final check fails.
How does the type decide the canonical form?
Two distinct real roots give two families of characteristics and the hyperbolic form u_vw equal to a lower-order expression. A repeated root gives one family, a free second coordinate and the parabolic form u_ww. Complex roots give no real characteristics, and real and imaginary parts are used as coordinates for an elliptic form.
What does d'Alembert's solution say physically?
Any solution of the wave equation on the whole line is the sum of one shape moving right and one moving left, each at speed c. With zero initial velocity, the initial profile splits into two halves of equal height that separate, and the value at a point depends only on data between the two characteristics through it.
Is the parabolic case harder than the hyperbolic one?
It has one fewer family of characteristics, so you choose the second coordinate yourself and must check the change of variables is invertible. The canonical form u_ww = 0 then integrates twice, giving one arbitrary function plus w times another. The extra choice is the step that markers check most carefully.
Exam move
Practise the three moves that every question in this chapter needs: compute B² − 4AC to fix the type, solve the auxiliary equation with its minus sign to get the characteristic slopes, then build the coordinates and integrate the canonical form. Do at least two hyperbolic examples with constant coefficients and one parabolic example, choosing the free coordinate so that the change of variables is clearly invertible.
For each, finish by substituting the general solution back into the original equation; the check is short because each arbitrary function only needs its second derivatives. For d'Alembert's solution, practise both the zero-velocity case, where the profile halves and splits, and a case with non-zero initial velocity, where you integrate the velocity data.
Sketch the two characteristics through a point to see its domain of dependence. Finally, link this week back to Week 6 by classifying each model equation again and to Week 7 by noting that each factor of the operator is a first-order characteristic problem. Keep a short table linking the sign of B² − 4AC, the number of real roots and the canonical form you expect.
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