Beskjeder
Corrected obligatory exercise set 2 will be handed out in the group lessons tomorrow.
NOTE: Both group lessons are in B1036, Nils Henrik Abels hus.
Group Lesson on Friday 29th: As many students have requested, we will have two different sessions of the special group lesson on tuesday.
The first session is in the morning (1015-1200) at B1036, Nils Henrik Abels Hus.
The second session is in the afternoon (1415-1600) at B1036, Nils Henrik Abels Hus.
You can attend either the morning or the afternoon session.
SYLLABUS FOR FINAL EXAM:
Heat equation: Solution by Fourier method, energy estimates and Maximum principles, Chapters 3.1 to 3.7, 6.2
Numerical methods for the Heat equation: Explicit and implicit finite difference schemes. Stability by Von-Neumann analysis and discrete maximum principles, Chapter 4.1,4.3,4,4,6.2.
Wave equation: Solution by Fourier Method, Energy estimates, Finite difference schemes, Numerical stability by Von-Neumann Analysis. Chapter 5.
SPECIAL Group lesson on 29th May, Friday: Review of the syllabus, solutions of previous years' question papers.
In today's lecture: I proved that the numerical scheme for approximating the Fisher equation is stable. The solutions of the equation were also proved to obey an invariant region principle and asymptotically converge to one.
Group lesson this friday: He will answer questions about oblig 2(if there are any).
NO lecture on next monday.
In today's lecture: I introduced a population model and solved the resulting ODE explicitly. The invariant region and asymptotic behavior of the ODE were described. The Fisher-Kolmogorov equation was also studied. A finite difference scheme was introduced.
In the next lecture: We find suitable conditions on the scheme to ensure an invariant region. An invariant region for the PDE and its asymptotic behavior will be studied.
Group lesson this friday: Prepare exercises 5.9, 5.11 and 11.1
Correction: No group lesson this friday on account of holiday.
In the lecture today: Trygve described an explicit finite difference scheme for the wave equation and showed that it is stable by using the Von-Neumann stability analysis.
In the group lesson this friday: Prepare exercises 5.9 and 5.11
Next lecture: I will describe a non-linear PDE that models population growth. The underlying ODE will be analysed and a finite difference scheme to approximate solutions will be designed. The material will be from chapter 11 of the book.
Today's lecture: Trygve obtained an explicit formula for the solution of the initial value problem for the wave equation. He also solved the boundary value problem using the method of separation of variables and represented the solution in terms of Fourier series. Uniqueness was established using energy method.
Group lesson this friday: Prepare exercises 5.1, 5.2, 5.3,5,5 and 5.8. Also show that results of 5.3(b) and 5.8(b) imply uniqueness of solutions.
Next lecture: Trygve will describe an explicit finite difference scheme to solve the wave equation numerically. Stability will be shown using Von-Neumann analysis.
Lecture on Monday 20th April: Trgyve Karper (CMA) will be giving the lecture. His email id is
trygvekk#math.uio.no
Replace # with @
He will start with initial value problem for the wave equation and then solve the initial boundary value problem for the wave equation with the Fourier method.
Group lesson this friday: Ulrik will solve some problems from the first obligatory exercise set.
The marks in the mid-term exam can be downloaded from
Group lesson this friday: He will solve the mid-term exam question paper
No classes on 6th and 13th due to Easter holidays.
Previous year question papers can be downloaded from
In Today's lecture: I described implicit schemes for the heat equation and proved that the scheme is unconditionally Von-Neumann stable.
SPECIAL GROUP LESSON THIS FRIDAY: I will revise the syllabus for the mid-term exam this friday and answer questions.
Syllabus for Mid-term Exam: ODEs and Numerical methods for ODE (1.2, 1.3), Solving first-order PDEs by method of characteristics (1.4.1, 1.4.2), Poisson's equation -- Green's function, Numerical schemes (2.1, 2.2, 2.3.4, 2.3.5), Heat equation -- Solution by Fourier series (3.1 to 3.6), Energy methods (3.7), Maximum principles (6.2.1,6.2.2)
Previous question papers: v07 midterm: folk.uio.no/siddharm/INF-MAT3360v07mtENG.pdf v08 midterm: folk.uio.no/siddharm/INF-MAT3360v08mtENG.pdf v07 final : folk.uio.no/siddharm/INF-MAT3360v07finalENG.pdf v08 final : folk.uio.no/siddharm/INF-MAT3360v08finalENG.pdf
In Today's lecture: I covered explicit schemes for the heat equation and proved Von-Neumann stability.
In the next lecture: I will desribe implicit schemes and prove stability by using Von-Neumann Analysis and the maximum principle.
For group lesson this friday: Prepare exercises 4.6, 4.8(a,c), 4.10 (a), 4.13, 4.18 (a,b,c).
In today's lecture: I derived maximum principles (chapter 6 of the book) for the heat equation and established some estimates based on them.
In the next lecture: I will describe explicit schemes for the heat equation and consider stability by using Von Neumann analysis.
For group lesson this friday: Prepare exercises 6.6 (a), 6.7 (a), 6.8 (a,b) and 6.9 (a).
In yesterday's lecture: I derived explicit formulas for the heat equation with Neumann boundary conditions. I also proved the energy estimate for the heat equation and used it to show uniqueness of solutions.
In the next lecture: I will derive stability and uniqueness by using maximum principles. I will describe explicit finite difference schemes for the heat equation.
For the group lesson this friday: Prepare exercises 3.5, 3.15, 3.16, 3.17, 3.18 and 3.20
The first obligatory exercise set can be downloaded from
folk.uio.no/siddharm/v09Oblig1.pdf
The due date is March 23rd.
In today's lecture: I obtained explicit solutions of the heat equation with the Fourier method.
In next lecture: I will cover the Neumann problem and derive stability estimates in terms of energy method and maximum principles.
For group lesson this friday: Prepare exercises 3.1, 3.4, 3,8, 3.11, 3.12 and 3.13.
In today's lecture: I proved stability and consistency estimates for the discrete approximations of the Poisson's equation and used them to show convergence of the discrete solutions at a quadratic rate.
In next lecture: I will consider the heat equation and construct solutions by using the Fourier method.
In the group lesson this friday: No specific exercises !! to prepare. Ulrik will be there to clear any doubts from chapter on Poisson's equations.
In today's lecture: I derived a finite difference scheme for the solutions of the Poisson's equation in one-dimension.
In next lecture: I will prove that the solution of this finite difference scheme converges to the solution of the Poisson's equation as the mesh is refined. I will also start with the solutions of the heat equation.
In the group lesson this friday: Prepare exercises 2.9, 2.12, 2.15, 2.16 and 2.17
In today's lecture: I solved the Poisson's equation in one-dimension explicitly in terms of the Green's function and derived some properties of the solution. I also derived uniquness of solutions by using Integration by parts.
In the next lecture: I will introduce finite difference approximations of the Poisson's equation and show that the approximate solutions are stable and convergent.
Group lesson this friday: Prepare exercises 2.2 to 2.6 and 2.8
In yesterday's lecture: I completed the solutions of transport equations by using the method of characteristics. I derived the heat equation and the Poisson's equation from the first principles.
Next monday's lecture: I will derive explicit solutions of the 1-D Poisson's equations in terms of the Green's function and derive some qualitative properties of the solution.
Group lesson this friday: Prepare exercises 1.5 (b,c,d) and 1.7. He will also explain the Gaussian elimination algorithm for solving systems of linear equations.
Scanned copies of chapter 2 from the book can be downloaded from
folk.uio.no/siddharm/kap2.pdf
Scanned copies of chapter 1 can be downloaded from
folk.uio.no/siddharm/chapter1.pdf