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Partial differential equations

Module overview

Differential equations occupy a central role in mathematics, as it enables us to describe a variety of real-world systems. The module aims to emphasize the importance of theory and applications of differential equations.

The module starts with the resumption of a part of the material from the first year module on differential equations, which draw attention to limiting value problems and also to equations with a source period. See you then how to express a general periodic function with regard to the Fourier series of sinus and cosine functions.

The second section of the module introduces some of the basic concepts of partial differential equations (PDES). It is shown how PDEs can be used to model situations in a variety of situations, including biology, finance and applied mathematics. The three important classes of the second order PDE suitable for modeling various types of phenomena are introduced, and the corresponding boundary conditions for each of these are taken into account. The technique of separation of variables is used to reduce the problem to solve the type of ordinary differential equations seen at the beginning of the module, and the general solution with Fourier series writes. During the entire module, it is a strong focus on problem solving and examples.

Linked modules

Requirements (math1056 or math1059 and math1060) and math2039 or (math1006 and math1007)

goals and goals

Learning results

knowledge and understanding

After successfully completing this module, you can demonstrate knowledge and understanding of:

.

  • be able to calculate the Fourier series
  • solve simple differential equations of the second order
  • understand the wide applications of the differential equation

Subject-specific intellectual and research capabilities

After successfully completing this module, you can:

  • Prove the orthogonality of intrinsic functions of limit value problems
  • be able to classify partially differential equations of the second order and select the corresponding boundary conditions

Subject-specific practical skills

After successfully completing this module, you can:

  • use laplace transformations to solve simple linear differential equations
  • Apply the method of separating variables to standard PDES

syllabus

Material Revision of material from the variation of parameters, green functions, limit issues,

Eigenvalues ??and ienfunctions,

Orthogonality of own functions

odd and even functions, half-range series,

Introduction to PDE's

PDE type of PDE: Hyperbolic, parabolic and elliptical and appropriate boundary conditions

Examples: Shaft Equations, Diffusion in Physical and Biological Systems, Black-Scholes Equation, Finance Applications, Potential Theory Applications in Biology and Physics.

Separation of variables

Separation of variables

Applications on diffusion problems in finance and biology

2-dim: harmonious functions

3-dim: basic solution

Definition and basic theory

Examples of transformations

transformations of derivatives

Applications to Ode and Systems

Applications to PDE's

Learning and teaching

Teaching and learning methods

Lectures, Operations, Private Study

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Recommendation

For example, we assess them if they do not meet the criteria to pass this module.

Tags: service, essay, course, work, excellent, equations, differential

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