Partial Differential Equations

Classification of second order equations in two independent variables. Problems with boundary conditions and/or initial conditions. Well-posed problems. Examples. Separation of variables and the problem of heat conductivity in a finite beam. Fourier series in sine and cosine and in complex form. Punctual convergence. Relationship between Fourier’s differentiability and transformation. Uniform convergence. Applications: heat conduction in a beam, finite vibrant string, and Dirichlet problem in the rectangle. Approximation by convolution and applications (Fejer’s theorem and Dirichlet’s problem on unitary disk). Periodic distributions. L²[ -π,π ] space, as subspace in periodic distributions. Notions of Hilbert spaces. Applications to the heat, wave, Poisson and Schrödinger equations. Additional suggested topics: Introduction to the Cauchy problem for the evolution of non-linear equations. Korteweg-de Vries equation in Sobolev spaces over the circle. Separation of variables and the problem of initial value for the heat in the plane equation. Fourier’s transformation in R. The inversion formula. Schwartz’ space and Fourier’s transformation. Tempered distributions. L²(R) spaces.

References:
EVANS, L. C. – Partial Differential Equations, Graduate Studies in Mathematics, 19, AMS, 1998.
FIGUEIREDO, D.G. – Análise de Fourier e Equações Diferenciais Parciais.  Rio de Janeiro, IMPA, Projeto Euclides, 1977.
GUSTAFSON, K. – Introduction to Partial Differential Equations and Hilbert space methods, New York, Wiley, 1980.
IÓRIO JR., R., IÓRIO, V. – Equações Diferenciais Parciais, uma introdução. Rio de Janeiro, IMPA, Projeto Euclides, 1988.
JOHN, F. – Partial Differential Equations, Springer-Verlag, New York, Third edition, 1978.
SOBOLEV, S. L. – Partial Differential Equations of Mathematical Physics. Reading Mass., Addison-Wesley, 1964.

* Standard program. The teacher has the autonomy to make any changes.