Advanced Mathematical Methods for Engineers (2015/2016)

Important Notice

• Classes will start on Monday September 28th.

• Ugo Gianazza

Office Hours

• Thursday from 4 pm to 6 pm and by appointment

Textbooks and Suggested Books

• Ordinary Differential Equations and Systems
• E.A. Coddington, An Introduction to Ordinary Differential Equations, Dover Publications, Inc., New York, 1961.
• M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, 1974.
• V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Differential Equations, Dover Publications, Inc., New York, 1989.
• W.T. Reid, Sturmian Theory for Ordinary Differential Equations, Applied Mathematics Series 31, Springer-Verlag, New York Heidelberg Berlin, 1980.

• Lebesgue Integral and Basic Tools of Functional Analysis
• E. DiBenedetto, Real Analysis, Birkhauser, Boston, (2002): Chapters III and V.
• B. D. Reddy, Introductory Functional Analysis, Texts in Applied Mathematics n. 27, Springer Verlag, New York, (1998).
• W. Rudin, Functional Analysis, Mc Graw Hill, New York, (1973).
• W. Rudin, Real and Complex Analysis, Mc Graw Hill, New York, (1966).
• A. Vasy, Partial Differential Equations: an Accessible Route through Theory and Applications, Graduate Studies in Mathematics, volume 169, American Mathematical Society, (2015): Chapters 1 and 13.

• Distributions
• E. DiBenedetto, Real Analysis, Birkhauser, Boston, (2002): Chapter VII.
• F.G. Friedlander, Introduction to the theory of distributions, Cambridge University Press, Cambridge, (1998).
• S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, 2nd Edition, Springer-Verlag Italia, (2015): Chapter 7.
• A. Vasy, Partial Differential Equations: an Accessible Route through Theory and Applications, Graduate Studies in Mathematics, volume 169, American Mathematical Society, (2015): Chapters 5 and 9.

• Partial Differential Equations
• E. DiBenedetto, Partial Differential Equations, 2nd Edition, Birkhaüser, (2009): Chapter 6.
• S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, 2nd Edition, Springer-Verlag Italia, (2015): Chapter 5.
• A. Vasy, Partial Differential Equations: an Accessible Route through Theory and Applications, Graduate Studies in Mathematics, volume 169, American Mathematical Society, (2015): Chapters 6, 7, 9, 11 and 12.

• Notes of the course Modelli e Metodi Matematici I (unfortunately in Italian)

Program

• Ordinary Differential Equations and Systems - Ordinary differential equations: in normal (or explicit) form, linear; order of a differential equation. Definition of solution. General solution and particular solution. General solution to the linear equation $y'(x) = \varphi(x)y(x) + \psi(x)$. General solution to a separable differential equation $y'(x) = X(x)Y(y(x))$. General solution to a homogeneous equation $y'(x)=f\left(\frac{y}{x}\right)$. The Cauchy Problem for equations and systems in normal form. Peano's local existence theorem. The local existence and uniqueness theorem. The global existence and uniqueness theorem. An extension theorem. The regularity theorem. Stability of solutions with respect to initial conditions and to parameters. Linear systems and equations of order $n$ with continuous coefficients: structure of the general solution. Liouville's Theorem (with proof). The method of variation of constants to determine a particular solution to a full, linear system (with proof). Linear constant-coefficient systems and equations. Homogeneous and general Boundary Value Problems for second order linear equations.
• Basic Tools of Functional Analysis - A short introduction to the Lebesgue integral. Normed spaces. Examples. Distance defined in terms of the norm. Equivalent norms. Convergent sequences. Cauchy sequences. Completeness, Banach spaces and their characterisation in terms of convergent series. Space with inner product. Main examples. Cauchy-Schwarz inequality. Continuity of the norm and of the inner product. Pythagoras' Theorem. Orthogonal Complement. Examples of spaces with inner product. Hilbert spaces. Projection Theorem. Meaning and applications; Orthonormal Systems. Fischer-Riesz's Theorem. Complete Orthonormal systems. Fourier Expansion. Parseval inequality. Gram-Schmidt orthonormalization process and applications. Simple examples of complete orthonormal systems in $L^2$. Best possible approximation in Hilbert spaces. Introduction to linear operators in normed spaces. Extension and restriction of a linear operator. Boundedness and continuity of a linear operator. Norm of a continuous operator. Continuity of the inverse operator. Continuity of linear operators in finite dimensional Hilbert spaces. Riesz representation theorem. Norm of a functional in a Hilbert Space; Representation of the Dirac delta in $H^1$. Adjoint of an operator in a Hilbert space; Symmetric operator; Self-adjoint operator. Applications of the theory of adjoint operators to Boundary Value Problems in Hilbert spaces; Eigenvalues and eigenvectors; Introduction to Sturm-Liouville Problems. Regular Sturm-Liouville problem. Examples of Sturm-Liouville problems which are not regular; Applications to Boundary Value Problems for complete equations.
• Distributions - Introduction to the Theory of Distributions. Definition of test function. Definition of distribution. Distributions and functions in $L^1_{loc}$. Definition of measure. Convergence in the sense of distributions. Derivatives in the sense of distributions. Fundamental solution of the Laplace equation in $\mathbb R^3$. Schwarz's Theorem in the framework of distributions. Vector-valued distributions. Div, grad, laplacian of a distribution. Product of a distribution and a function. Composition of a distribution with a function. Tensor product of distributions. Applications of the composition formula. Restriction of a distribution. The problem of division in the framework of distributions: homogenous and complete case. Convolution of a distribution with a function. Main properties of the convolution of a distribution with a function. Space of rapidly decreasing functions. Notion of convergence in $\mathcal{S}(\mathbb R^N)$. Definition of tempered distributions. Notion of convergence for tempered distributions. Fourier transform for tempered distributions. Simple Examples. Main properties of the Fourier transform. Convolution between a distribution and a function. Convolution between two different distributions. Convolution Theorem. $H^k(\Omega)$ spaces. Characterization of $H^k(\mathbb R^N)$ in terms of the Fourier transform. Paley-Wiener Theorem. Application of Fourier transform methods to ordinary differential equations.
• Partial Differential Equations - Introduction to the Theory of Partial Differential Equations. Waves. Derivation of the $1$-dimensional wave equation under simplifying assumptions. Conservation of the energy. Initial and Boundary Value Problems for the $1$-dimensional wave equation; Existence, uniqueness and continuous dependence with respect to the data of the solution for a IBVP constructed using the separation-of-variable method. Global solution to the homogeneous $1$-dimensional wave equation: existence, uniqueness and stability. Domain of dependence. Regularity of the solution. Generalised solutions. Duhamel's method for the complete $1$-dimensional wave equation. Plane waves in $\mathbb R^N$; Cylindrical and spherical waves in $\mathbb R^3$. Existence of a solution for the global Cauchy Problem in $\mathbb R^N$. Definition of fundamental solution. Fourier transform of the fundamental solution with respect to the space variable. Uniqueness of the solution to the global Cauchy Problem in $\mathbb R^N$. Uniqueness of the solution to initiali-boundary value problems in regular domains. Existence of a solution for the bidimensional square membrane. Coincidence between the solution to the global Cauchy Problem and the distributional solution of a proper complete wave equation. Explicit expression of the $1$-dimensional wave equation. Support of the fundamental solution to the wave equation. Special features of the $3$-dimensional wave equation. Fundamental solution of the $3$-dimensional wave equation. Main properties of the solution. Stability in $L^\infty(\mathbb R^3)$ of the unique solution. Fundamental solution to the $2$-dimensional wave equation. Explicit formulation of the unique solution to the global Cauchy Problem. Main properties of the solution. Duhamel's method for the solution of the complete wave equation: $3$-dimensional and $2$-dimensional cases. Solution of the $3$-dimensional wave equation with a point source and with a moving source: comments on the general case, and explicit resolution both of the subsonic and of the supersonic regimes.

Final Exam

• The final exam consists in a written test and an oral exam.

Written Test Schedule

• February 2nd 2016, at 9.00 am, classroom EF1
• February 22nd 2016, at 9.00 am, classroom EF1
• April 5th 2016, at 9.00 am, classroom E7 (restricted)
• June 23rd 2016, at 9.00 am, classroom EF4
• July 7th 2016
• August 30th 2016
• September 15th 2016

Some exercises (some of them in Italian)

• Qualitative studies of solutions to first order differential equations
• Consider the differential equation $$y'=1+\arctan y^2.$$
• Prove that the general solution is defined on $\mathbb R$ and it is of $C^\infty$ class.
• Draw a qualitative graph of the solutions.
• Consider the differential equation $$y'={y^2-x^2\over{1+y^2}}.$$
• Determine the set where the general solution is defined.
• Study the sign of the first order derivative in $\mathbb R^2$ and characterize the set where $y'$ vanishes.
• Relying on the results of the previous issues, draw a qualitative graph of the solution which satisfy the initial condition $y(-1)=-1$, $y(1)=1$, $y(0)=0$.
• Draw qualitative graphs of the particular solutions to the equation $$y'=(y^2-1)e^{{y^2}}.$$
• Draw qualitative graphs of the particular solutions to the equation $$y'=y(1-e^{{x^2}}).$$
• Draw qualitative graphs of the particular solutions to the equation $$y'={y(y-1)\over{1+y^2}}.$$
• Consider the Cauchy Problem \begin{equation*} \left\{ \begin{aligned} & y'_\alpha=y_\alpha\sin(e^x+y_\alpha)-e^x\\ & y_\alpha(0)=\alpha\qquad\qquad\alpha\in{\mathbb R}. \end{aligned} \right. \end{equation*} Prove that $\forall\alpha\in{\mathbb R}$ there exists a unique $y_\alpha:{\mathbb R}\rightarrow{\mathbb R}$ solution to the problem, which is of $C^\infty$ class and draw a qualitative graph of $y_\alpha$ when $\alpha\in(-1-2\pi,-1-\pi)$.
• Consider the Cauchy Problem \begin{equation*} \left\{ \begin{aligned} & y'=y(y+x)e^{-|y|}\\ & y(0)=1 \end{aligned} \right. \end{equation*} and draw a qualitative of the solution and show that it is of $C^\infty$ class.
• Consider the differential equation $$y'=x(\exp({{1\over{y^2}}})-e).$$
• Determine the set $\Omega\subseteq{\mathbb R}^2$ where it is possible to ensure the existence and uniqueness of the solution to the Cauchy Problem given in $(x_o,y_o)\in\Omega$.
• Draw qualitative graphs of the solutions, assuming that the number of inflection points is as small as possible.
• Relying on the previous results, determine for which values of $\lambda\in{\mathbb R}$ the solution to the initial condition $y(0)=\lambda$ can be extended on ${\mathbb R}$ and for which values of $\lambda$ it is bounded as well.
• Exercises on First Order Differential Equation (in Italian) pdf;
• Exercises on Linear Constant-Coefficient Differential Systems (in Italian) pdf;
• Exercises on Linear Constant-Coefficient Differential Equations (in Italian) pdf;
• Exercises on the Lebesgue Integral
• Compute $$\lim_{n\to+\infty}\int_n^{n^2+1}\sin(nx)\,\exp\left(\left(\frac1n-2\right)x^3\right)\,dx.$$
• Compute $$\lim_{n\to+\infty}\iint_{\mathbb R^2}(y e^{-y})_+ (x^2-2x-1)_-[1-\exp(-n\sin^2 x)]dxdy,$$ where for a given function $f$, we define $(f)_\pm:=\max\{\pm f;0\}.$
• Relying on the Dominated Convergence Theorem, compute $$\lim_{n\to+\infty}\int_0^{+\infty}\frac{n}{3x}\arctan\left(\frac{4x}{n}\right)\frac1{16+x^2}dx.$$
• Exercises on Normed Spaces
• Consider the sequence of functions $f_n:{\mathbb R}\to{\mathbb R}_+$ with $n\ge1$ defined by $$f_n(x)=|\cos x|^{\frac1n}\,e^{-n|x|}.$$
• Study the pointwise limit of the sequence.
• Prove that $\{f_n\}\subset L^1({\mathbb R})$.
• Prove that $\{f_n\}\subset L^2({\mathbb R})$.
• Prove that $\{f_n\}\subset C^0_b({\mathbb R})$, the vector space of the bounded and continuous functions defined on $\mathbb R$.
• Study the limit of the sequence in $L^1(\mathbb R)$ endowed with its natural norm.
• Study the limit of the sequence in $L^2(\mathbb R)$ endowed with its natural norm.
• Study the limit of the sequence in $C^0_b(\mathbb R)$ endowed with the supremum norm.
• Consider the sequence $\{f_n\}$, $n\in{\mathbb N}$, where $$f_n(x)=2xe^{-3x/n}.$$
• Prove that $\{f_n\}\subset C^0([0,1])$, and $\{f_n\}\subset L^1(0,1)$.
• Compute the pointwise limit of $\{f_n\}$.
• Study the convergence of the sequence in $C^0([0,1])$ endowed with the supremum norm, and in $L^1(0,1)$ endowed with the integral norm.
• Consider the vector space $$C^1([0,1]=\left\{f:[0,1]\to{\mathbb R}:\ \ f\in C^0([0,1]),\ f'\in C^0([0,1])\right\}.$$ Discuss which of the following quantities is a norm:
• $\textstyle \sup_{x\in[0,1]}|f'(x)|$;
• $\textstyle \sup_{x\in[0,1]}|f(x)|+\sup_{[0,1]}|f'(x)|$;
• $\textstyle \int_0^1|f'(x)|\,dx$;
• $\textstyle \int_0^1 |f(x)|\,dx+\int_0^1|f'(x)|\,dx$.
• Consider the sequence $\{f_n\}$, $n\in{\mathbb N}$, where $$f_n(x)=\chi_{[n,n^2]}(x)\,\cos(2x)\,e^{-nx}.$$
• Prove that $\{f_n\}\subset L^1(0,1)$, and $\{f_n\}\subset L^2(0,1)$.
• Compute the pointwise limit of $\{f_n\}$.
• Study the convergence of the sequence in $L^1(0,1)$ endowed with the integral norm, and in $L^2(0,1)$ endowed with the integral norm of $|f|^2$.
• Exercises on Linear operators
• Check that the following operators are linear and bounded. Compute their norms.
• $A: C^0([0,1])\to C^0([0,1])$, $(Ax)(t):=t^2 x(0)$;
• $A:L^2(0,1)\to L^2(0,1)$, $(Ax)(t):=x(t)\chi_{(0,\frac12)}(t)$;
• $A:L^2(0,1)\to L^2(0,1)$, $(Ax)(t):=\int_0^t x(\tau)\,d\tau$.
• Let $A:C^0([0,1])\to C^0([0,1])$ defined by $(Ax)(t):=\int_0^t x(\tau)\,d\tau +x(t).$ Prove that ${\mathcal N}_{(A)}=0$, check tha the inverse operator is continuous, and compute it explicitly.
• Consider $K:[a_1,b_1]\times[a_2,b_2]\to \mathbb R$ with $K\in C^0([a_1,b_1]\times[a_2,b_2])$. Prove that the operator $A: C^0([a_2,b_2])\to C^0([a_1,b_1]),\qquad (Ax)(s):=\int_{a_2}^{b_2}K(s,t)x(t)\,dt$ is linear and continuous.
• Exercises on Distributions pdf;

 Last modification: September 15th 2016