This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines. The main purpose is on the one hand to train students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other hand to give them a solid theoretical background for numerical methods. At the end of each chapter, a number of exercises at different level of complexity is included

second order equations in divergence form. The last section contains an application to a simple control problem, with both distributed observation and control. The issue in chapter 9 is the variational formulation of evolution problems, in particular of initial-boundary value problems for second order parabolic operators in divergence form and for the wave equation. Also, an application to a simple control problem with ﬁnal observation and distributed control is discussed. At the end of each

variables method that we describe below through a simple example of heat conduction. We will come back to this method from a more general point of view in Section 6.9. As in the previous section, consider a bar (that we can consider one-dimensional) of length L, initially (at time t = 0) at constant temperature u0 . Thereafter, the end point x = 0 is kept at the same temperature while the other end x = L is kept at a constant temperature u1 > u0 . We want to know how the temperature evolves

solution of (2.20) is 2 wm (s) = Ce−m π2s (C arbitrary constant). (2.22) From (2.21) and (2.22) we obtain damped sinusoidal waves of the form 2 Um (y, s) = Am e−m π2 s sin mπy. Step 3. Although the solutions Um satisfy the homogeneous Dirichlet conditions, they do not match, in general, the initial condition U (y, 0) = y. As we already mentioned, we try to construct the correct solution superposing the Um by setting ∞ 2 Am e−m U (y, s) = π2s sin mπy. (2.23) m=1 Some questions

deﬁnition of a probability measure μ and of the integral with respect to the measure μ . 62 2 Diﬀusion 2.6.2 Walks with drift and reaction As in the 1−dimensional case, we can construct several variants of the symmetric random walk. For instance, we can allow a diﬀerent behavior along each direction, by choosing the space step hj depending on ej . As a consequence the limit process models an anisotropic motion, codiﬁed in the matrix ⎛ ⎞ D1 0 · · · 0 ⎜ 0 D2 0 ⎟ ⎜ ⎟ D =⎜. . . . . .. ⎟ ⎝ .. ⎠ 0

reduced to a global Cauchy problem for the heat equation. In this way it is possible to ﬁnd explicit formulas for the solutions. First of all we make a change of variables to reduce the Black-Scholes equation to constant coeﬃcients and to pass from backward to forward in time. Also note that 1/σ 2 can be considered an intrinsic reference time while the exercise price E gives a characteristic order of magnitude for S and V. Thus, 1/σ 2 and E can be used as rescaling factors to introduce