Introduction to PDEs
I was working on a different topic for the blog and as I’m kinda busy packing my things to depart for university and balancing uni classes and courses and studying PDE ( a concept I was immensely scared of), it suddenly came to me why was i even scared of it in the first place? So I decided to break it up into slabs of question and I feel it is helping me grasp the concept now(it seem *fingers crossed*) So i’ll share my review hope it helps!
How is ODE different from PDE?
When i had the prerquisite to the course earlier in my UG I always wondered why ODE wasn’t enough? Why did we need PDE on the first place?
Foolish me! The answer was right in front!
ODE: Ordinary Differential Equations are equations where derivatives are taken wrt only one variable.[Hence there will only be only one independent variable]
PDE: These are those equations that depend on partial derivatives of more than one(/several) variables.[Hence they possess derivatives of more than one variable]
Just like a Fierce Queen, mathematics also goes by her own rules(and makes more if situation demands, but no compromise). So we often come across some practical situations be it finding noise in communication systems , radioactive disintegration or just the flow of heat which demands us to find solution where ODE’s just aren’t sufficient. Hence PDE was introduced(in my opinion)
The modern notation of PDE was introduced by Adrien Marie Legendre in 1786.
Structure of PDE
We denote a PDE generally like this:
Conventionally we use:
Hence fig-1 can also be expressed as: F(x,y,z,p,q)=0 [fig-3]
Now we need to be aware that the solution of eq 1 in some domain D of R² is a function z=f(x,y). It satisfies the following conditions:
- For every (x,y) belongs to D, the point (x,y,z,p,q) is the domain of the function F.
- When we substitute z=f(x,y) into eqution 1(ie fig-1), it reduces toa n identity in x,y for all (x,y) belonging to domain D.
Order & Degree:
Order: The order of the PDE is the order of the highest order partial derivative in the equation or PDE.
Degree: The degree of the PDE is the degree of the highest order partial derivative in the equation after it is rationalized(i.e free from all radicals & fractions) as far as derivatives are concerned.
Classifications:
PDEs can be classified depending on the form of function F.
- QUASILINEAR PDE: The PDE is said to be quasilinear when x,y are independent variables and z=z(x,y) is dependent and is of form P(x,y,z)p+Q(x,y,z)q=R(x,y,z)
- LINEAR PDE: A PDE is said to be linear if the dependent variable and all its derivatives are in first order and the dependent variable and/or its derivatives are not in product form.
- NON-LINEAR PDE: A PDE is which is not linear is called non-linear PDE.
Well this is it for today, If there is something you want me to discuss on please feel free to use the comment section and let me know!
I usually follow:
An Introduction to DE by Ghosh & Maity
Ordinary & Partial DEs by Raisinghania
An introduction to PDEs by K. Sankara Rao
Thank You Guys! See you next time!
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