Exam Questions PhD
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PhD – 2010 Mathematical Physics Questions Section 1 STRING VIBRATION 1 Formula for the general solution of the string vibration equation. 2 Solution of Cauchy problem for the non homogeneous unbounded string vibration equation. 3 On the uniqueness of the solution of Cauchy problem for unbounded string. 4 On the stability of the Cauchy problem for the string vibration. 5 Solution of Mixed problem for the half bounded string. Method of continuation. First boundary conditions. 6 Solution of Mixed problem for the half bounded string. Method of continuation. Second boundary conditions. 7 Bounded string. Method of separation of variables. Section 2 HEAT CONDUCTION 1 Maximum principles. 2 Uniqueness of the solution of the boundary problems for the heat conduction in a bounded rod. 3 Stability of the solution of the boundary problems for the heat conduction in a bounded rod. 4 Solution of the 1-st boundary problem for heat conduction in a bounded rod. 5 The problem on Linear distribution of heat. 6 Theorem on uniqueness of solution of the Cauchy problem for heat conduction in unbounded rod. 7 Solution of Cauchy problem for heat conduction equation: Unbounded rod. 8 Fundamental solution of heat equation and its properties. 9 Solution of 1-st boundary problem for heat conduction in a half bounded rod for the non homogeneous equation. Section 3 ELLIPTIC EQUATIONS. 1 Fundamental solutions of Laplace equation in a
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Exam Questions PhD Exam Questions PhD Exam Questions PhD
Transcript of Exam Questions PhD
METRIC SPACES, Page 37
PhD 2010Mathematical Physics
QuestionsSection 1String vibration
1Formula for the general solution of the string vibration equation.
2Solution of Cauchy problem for the non homogeneous unbounded string vibration equation.
3On the uniqueness of the solution of Cauchy problem for unbounded string.
4On the stability of the Cauchy problem for the string vibration.
5Solution of Mixed problem for the half bounded string. Method of continuation. First boundary conditions.
6Solution of Mixed problem for the half bounded string. Method of continuation. Second boundary conditions.
7Bounded string. Method of separation of variables.
Section 2Heat Conduction
1Maximum principles.
2Uniqueness of the solution of the boundary problems for the heat conduction in a bounded rod.
3Stability of the solution of the boundary problems for the heat conduction in a bounded rod.
4Solution of the 1-st boundary problem for heat conduction in a bounded rod.
5The problem on Linear distribution of heat.
6Theorem on uniqueness of solution of the Cauchy problem for heat conduction in unbounded rod.
7Solution of Cauchy problem for heat conduction equation: Unbounded rod.
8Fundamental solution of heat equation and its properties.
9Solution of 1-st boundary problem for heat conduction in a half bounded rod for the non homogeneous equation.
Section 3Elliptic equations.
1Fundamental solutions of Laplace equation in a plane.
2Fundamental solutions of Laplace equation in a space.
3Greens formulas 1-st and 2-d.
4Main Greens formula.
5Mean Value theorem for harmonic functions.
6Maximum principle for harmonic functions.
7Uniqueness of the solution of Direchlets problem for Poissons equation.
8Stability of the solution of Direchlets problem for Poissons equation.
9Laplaces equation in curvilinear coordinates: Polar coordinates.
10Separation variables for the 1-st boundary value internal problem in a circle.
11Separation variables for the 1-st boundary value external problem in a circle.
Section 4: Additional chapters
1.Basic Problems of Mathematical Physics. Un bounded domains. 2 dimension.
2.Bounded domains. Mixed Problems. 2- dimension.
3.Fourier Method. Sturm Liouville theory. 2-dimention problems.
4.Trigonometric Fourier series. Some basic notions and theorems.
5.Hilbert spaces. Fourier series.
6.Problems in unbounded domains. Fourier transformation. Fourier Integrals. N- dimension.
7.Multiple Fourier series and Integrals. Summability.
8.Mathematical Physics problems in a unit ball. Fourier-Laplace series on a sphere.
9.Spaces of Test functions.
10.Distributions.
Rules:
It will be 8 questions 2 from each section. Total 100.
