MA5102 : Partial Differential Equations (Autumn 2021)

Details of MA5102 (Autumn 2021)

Level: 5

Type: Theory

Credits: 4.0

Course Code	Course Name	Instructor(s)
MA5102	Partial Differential Equations	Rajib Dutta, Saugata Bandyopadhyay

Syllabus
First-order Equations: Method of characteristics and existence of local solutions. Characteristic Manifolds and Cauchy Problem: Non-characteristic surfaces, Cauchy-Kowalevski theorem and uniqueness theorem of Holmgren. Laplace Equation: Fundamental solution, harmonic function and its properties, Poissons equation, Dirichlet problem and Greens function, existence of solution of the Dirichlet problem using Perrons method, introduction to variational method. Heat Equation: Fundamental solution and initial-value problem, mean value formula, maximum principle, uniqueness and regularity, nonnegative solutions, Fourier transform methods. Wave Equation: dAlemberts formula, method of spherical means, Hadamards method of descent, Dumahels principle and Cauchy problem, initial-boundary-value problem, Fourier transform methods.

Syllabus

First-order Equations: Method of characteristics and existence of local solutions.

Characteristic Manifolds and Cauchy Problem: Non-characteristic surfaces, Cauchy-Kowalevski theorem and uniqueness theorem of Holmgren.

Laplace Equation: Fundamental solution, harmonic function and its properties, Poissons equation, Dirichlet problem and Greens function, existence of solution of the Dirichlet problem using Perrons method, introduction to variational method.

Heat Equation: Fundamental solution and initial-value problem, mean value formula, maximum principle, uniqueness and regularity, nonnegative solutions, Fourier transform methods.

Wave Equation: dAlemberts formula, method of spherical means, Hadamards method of descent, Dumahels principle and Cauchy problem, initial-boundary-value problem, Fourier transform methods.

Prerequisite
Ordinary Differential Equations (MA4202) and Fourier Analysis (MA4205)

References
Suggested Texts: Evans, L.C., Partial Differential Equations, American Mathematical Society. Folland, G., Introduction to Partial Differential Equations, Princeton University Press. Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order, Springer-Verlag. Han, Q., A Basic Course in Partial Differential Equations, American Mathematical Society. John, F., Partial Differential Equations, Springer-Verlag. McOwen, R.C., Partial Differential Equations: Methods and Applications, Pearson Education. Renardy, M. and Rogers, R., An Introduction to Partial Differential Equations, Springer-Verlag.

References

Suggested Texts:

Evans, L.C., Partial Differential Equations, American Mathematical Society.
Folland, G., Introduction to Partial Differential Equations, Princeton University Press.
Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order, Springer-Verlag.
Han, Q., A Basic Course in Partial Differential Equations, American Mathematical Society.
John, F., Partial Differential Equations, Springer-Verlag.
McOwen, R.C., Partial Differential Equations: Methods and Applications, Pearson Education.
Renardy, M. and Rogers, R., An Introduction to Partial Differential Equations, Springer-Verlag.

Course Credit Options

Sl. No.	Programme	Semester No	Course Choice
1	IP	1	Not Allowed
2	IP	3	Not Allowed
3	IP	5	Not Allowed
4	MR	1	Not Allowed
5	MR	3	Not Allowed
6	MS	3	Not Allowed
7	MS	5	Not Allowed
8	MS	7	Not Allowed
9	MS	9	Elective
10	RS	1	Elective
11	RS	2	Elective

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Details of MA5102 (Autumn 2021)

Course Credit Options