MTH647 Midterm Handouts

Mathematical Modelling: It is the approximation of objects with geometrical objects. Mathematical Formulation: In this step we drive the equation corresponds to the given phenomenon. It is based on existing law i.e. Newton’s law of motion, laws of thermodynamics. If laws are not available for given existing problem, then we go for experiments. Example:

Example: Heat flow problem Formulation PDEs Solutions Expansion of function terms of ‘sine’ and ‘cosine’ (Fourier analysis). Physical Interpretation: E.g. dx x dt t x t Ae A R

Vibrating String Equation:

y y a t x Here we’ll try to derive the above given equation. So, Let represents the elements of arc of string and is tension at and (constant).

For small angles; So …… (1) By newton second law of motion

…… (2) Is linear mass density Note: Here acceleration is in derivative form and while discussing the acceleration for a segment (large number of particles), there will be some approximation/error. That’s why we are using as approximation symbol. By comparing equation 1 and 2, we have

As , so above equation become

Note: Here are respectively. For small vibrations; By using this relation, we get

Dividing by on both sides

For and , it becomes

Here , so equation becomes

Or it can be written as

This relation is known as Vibrating string equation.

Boundary Conditions for Vibrating String Equation:

Statement: Write boundary conditions for a vibrating string of length for which i) Ends and are fixed. ii) Initial shape of string is . iii) Initial velocity distribution is . iv) Displacement at any instant is bounded.

Solution:

i) String is fixed at and , then its displacement at

Note: above relation is in form. At starting point and at final point . ii) Initial shape at is given as

Note: Just put in relation and here is just an expression. iii) Initial velocity at time is given as

At Note: Here velocity is in derivative form and is just an expression. iv) Displacement is bounded: It implies that their exist some finite real number such that

These are some examples of boundary conditions applied on vibrating string equation.

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Heat Flux across the Plane:

And Heat flows from higher temp plane to lower temp plane. Heat flux is defined as; Heat per unit area per unit time. Mathematically

∆𝑛 ∆𝑛 Where ′𝑘′ is constant of proportionality and is called thermal conductivity. Thermal conductivity is defined as; Measure of how well a material can conduct or transfer heat.

negative value of heat flux. (higher temp to low temp) and vice versa. For limiting case: If then

𝜕𝑛 In vector form ∇n

Partial Differential Equations (Definition and Related Terms):

Definition: It is an equation containing unknown functions of two or more variables and partial derivatives w.r.to these variables. Example 𝜕2𝑢

Order: It is the order of highest derivative involved in PDE. Solution: It is a function which satisfy the given DE. Example:

𝜕𝑢

𝜕2𝑢

Particular Solution: It is obtained from the general solution by particular choice of arbitrary functions. Example:

Singular Solution: It cannot be obtained from the general solution by choosing arbitrary functions.

Example: (General Solution−Here ′𝑐′ is arbitrary constant). But ′𝑦= 0′is also a solution of given DE.As it is not obtained from general solution, that’s why it is called