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Questions and Answers
What is the basis of mathematical formulation?
What is the basis of mathematical formulation?
What is mathematical modelling?
What is mathematical modelling?
In the context of the vibrating string equation, what does 'y' represent?
In the context of the vibrating string equation, what does 'y' represent?
What does 'dx/dt' represent in the context of the vibrating string equation?
What does 'dx/dt' represent in the context of the vibrating string equation?
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What is the purpose of Fourier analysis in the context of the heat flow problem?
What is the purpose of Fourier analysis in the context of the heat flow problem?
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In the context of vibrating string equations, what does the boundary condition 'displacement is bounded' imply?
In the context of vibrating string equations, what does the boundary condition 'displacement is bounded' imply?
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What is the definition of heat flux?
What is the definition of heat flux?
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What does a negative value of heat flux indicate in the context of thermal conductivity?
What does a negative value of heat flux indicate in the context of thermal conductivity?
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What does a limit case where ∆𝑛 approaches 0 and ∆𝑢 approaches 0 imply for heat flux?
What does a limit case where ∆𝑛 approaches 0 and ∆𝑢 approaches 0 imply for heat flux?
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What is a Partial Differential Equation (PDE) by definition?
What is a Partial Differential Equation (PDE) by definition?
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Study Notes
Mathematical Formulation
- Mathematical formulation involves using mathematical language to represent real-world problems and phenomena. It relies on defining variables, relationships, and equations to capture the essence of a problem.
- Example: The vibrating string equation, which expresses the relationship between the displacement of a string, its time derivative, and its second spatial derivative, provides a mathematical representation of how a string vibrates.
Mathematical Modelling
- Mathematical modelling is the process of creating simplified representations of real-world situations using mathematical tools. These models usually involve simplifying assumptions and neglecting some aspects of reality.
- Example: The vibrating string equation is a simplified model of the actual string vibration because it omits the complexities of the string material and external forces.
Vibrating String Equation: y
- Represents the displacement of a string at a specific point (x) along its length, at a given time (t).
Vibrating String Equation: dx/dt
- Represents the rate of change of displacement of the string at a specific point (x) along its length, with respect to time (t). In other words, it's the velocity of the string at that point.
Fourier Analysis in Heat Flow
- Fourier analysis uses a series of sine and cosine functions to represent complex, periodic functions.
- In the context of the heat flow problem, Fourier analysis breaks down complicated temperature profiles into simpler waves, allowing for a clearer understanding of the heat transfer process.
Boundary Condition: Displacement Bounded in Vibrating Strings
- Implies that the displacement of the string at the ends is limited. This condition is usually applied to represent fixed ends of the string, where the displacement cannot exceed certain limits.
Heat Flux Definition
- It's the rate of heat energy transfer per unit area, meaning the amount of heat energy passing through a specific area per unit time.
Negative Heat Flux
- Indicates heat flow in the opposite direction to the assumed positive direction. For example, a negative flux value might represent heat flowing from a warmer region to a colder region.
Limit Case of ∆n and ∆u Approaching 0
- Implies that heat flux is being calculated at an infinitesimally small area and temperature difference. This limit case represents a more precise calculation of heat transfer at a particular point.
Partial Differential Equation (PDE) Definition
- A partial differential equation (PDE) is a mathematical equation involving unknown functions of several independent variables and their partial derivatives. It describes the relationship between these functions and their changes with respect to multiple variables.
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Description
Learn about mathematical modelling and mathematical formulation in the context of mathematical physics with this lecture handout for MTH647 at Virtual University of Pakistan. The lecture covers the approximation of objects with geometrical objects and the derivation of equations corresponding to physical phenomena.
