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Munson, Donald F. Young, Theodore H. Incropera D. Graham Solomons Fundamentals of Physics 7th Ed.

Homogeneous Differential Equations

Petrucci; William S. Lang and G. Some techniques on ordinary differential equations as seen in MA Differential Equations , on uniform convergence of series as taught in MA Analysis III , and on the divergence theorem as presented in MA Multivariable Calculus will be needed and only briefly introduced in the lectures. Content : The theory of partial differential equations PDE is important both in pure and applied mathematics.

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On the one hand they are used to mathematically formulate many phenomena from the natural sciences electromagnetism, Maxwell's equations or social sciences financial markets, Black-Scholes model. On the other hand since the pioneering work on surfaces and manifolds by Gauss and Riemann partial differential equations have been at the centre of many important developments on other areas of mathematics geometry, Poincare-conjecture.

Subject of the module are four significant partial differential equations PDEs which feature as basic components in many applications: The transport equation, the wave equation, the heat equation, and the Laplace equation.

Midterm Exam

We will discuss the qualitative behaviour of solutions and, thus, be able to classify the most important partial differential equations into elliptic, parabolic, and hyperbolic type. Possible initial and boundary conditions and their impact on the solutions will be investigated. Solution techniques comprise the method of characteristics, Green's functions, and Fourier series.

Aims : To introduce the basic phenomenology of partial differential equations and their solutions. To construct solutions using classical methods. Objectives : At the end, you will be familiar with the notion of well-posed PDE problems and have an idea what kind of initial or boundary conditions may be imposed for this purpose.

No books, notes, or calculators of any kind are allowed. Again, understanding derivations and proofs is important. Sturm-Liouville problem; self-adjoint operators; orthogonality and completeness of e-functions; reality and positivity of e-values of S-L problems; regular vs singular SL problems. Fredholm alternative 5. Boundary value problems in several dimensions: equations of Laplace, Poisson, Helmholtz; Dirichlet and Neumann problems in bounded domains.

Distribution theory: test functions; definition and properties of distributions; definition of a delta distribution and its sifting property; derivative of a distribution; a dipole distribution. Solution of the wave equation in the sense of distributions.


Fundamental solution of the Helmholtz equation. Fundamental solution of the heat equation. Hyperbolic systems of linear PDE; diagonalization.

Fourier transforms; basic properties; transform of distributions: Heaviside step function and delta distribution; convolution theorem; sine and cosine transforms. Rankine-Hugoniot conditions.


Lax entropy conditions. Debnath, Linear PDE for Scientists and Engineers Catalog description: First part of a sequence of courses on partial differential equations PDE emphasizing theory and solution techniques for linear equations.

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