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## Partial differential equations

A Partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives. They are used to formulate problems involving functions of several variables. They can either be solved by hand or used to create a relevant computer módel.

Almost all the elementary and numerous advanced parts of theoretical physics are formulated in terms of differential equations. Often partial differential equations.

The most frequently encountered are:

1. Laplace’s equation: 2   ψ = 0. Important in the study of electromagnetic phenomena, dielectrics, steady currents and magnetostatics, hydrodynamics (irrotational flow of a perfect fluid, and surface waves, heat flow, gravitation.)
2. Poisson’s equation:∇2   ψ = ρ/ε. Non-homogenous with a source term.
3. The Helmholtz equation or wave equation:

2   ψ+ k2ψ = 0 and time-independent diffusion equations. This equation can be used to describe : elastic waves in solids, bars, membranes, sound, acoustic waves, nuclear reactors

1. The time dependent diffusion equation: ψ= δψ/δta-2 .
2. The time-dependent wave-equation □2ψ =0.
3. The scalar potential equation. □2ψ = -ρ/ε.
4. The Klein-Gordon equation □2ψ =μψ
5. The Schrödinger equation

-h2/(2m)∇ψ + V ψ = Eψ describing the motion of the sub-atomic particles.

1. Maxwell’s coupled differential equationsfor electric and magnetic field and Dirac’s equation for the relativistic electron wave
2. function.
3. Differential form
Name ”Microscopic” equations ”Macroscopic” equations
Gauss’s law
Gauss’s law for magnetism
Ampère’s circuital law
(with Maxwell’s correction)
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## Differential equations of the second order

Second order differential equations of the homogen type

y” (x)+ a y'(x) + by(x) = 0

are possible to solve with the aid of the characteristic equation

r2 + a r +b =0

If this have the roots r1 and r2

the solution is given by

y(x) = Cer1x + Der2 x

If the equation is inhomogenous and the right side is a polynom assume a solution a polynomial of the same degree

If the right side is a trigonometric function assume a as a solution a combination of trigonometric functions.

Ex. Solve the equation y” -3y – 4y = 0

Solve the characteristic equation: r2-3r-4 = 0
r=4 och r = -1.

The general solution is y = Ce-4x + Dex
Where C and D are arbetare constants.

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## Similarity

Two triangles are similar if all the corresponding angles are equal.

Then the ratios of the corresponding sides to each other are the same for both triangles.

In the fig above AB/BC = DE/EF or AB/DE = BC/EF.

A list of all the cases of similarity:

Fractals are made up of similar geomtetrical objects.

See e.g. the Serpinski triangle.

Fracrtals can be used to describe objects encounterd in real life such as coastlines, trees or clouds.

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## How to draw exceldiagrams

My instruction video concerning drawing diagrams in Excel:

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## Separable variables

Differential equations of the form

dy/dx = – P(x)/Q(y)

then it is possible to separate the variables

Q(y)dy = – P(x) dx → Q(y) dy + P(x) dx = 0

Ex

y´+ sinx y = 0

y´ = -sinx y

dy/y = -sinx dx

Integrating both sides

lny = cosx +C

y = Decosx

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## Differential equations

An equation containing the derivative of a function is called a differential equation.

Depending on the order of the derivatives it is of the first, second or higher order.

The simplest differential equation is an ordinary linear homogenous differential equation of the first order:

y’ + 3y = 0.

The solution to this equation is given with the integrating factor: e-3x

Where the exponent is the primitive function for the coefficient in front of y.

Multiplication of both sides with this factor gives:

e-3x  y’ –  3y e-3x    = 0

The left side is identical to the derivative of the product D(y  e-3x ).

Therefore  integration of both sides generates y e3x  =C and y = C e-3x

which is the solution C being an arbitrary constant.

For inhomogenous equations the solution is given by the sum of the solution to the homogenous equation and a particular solution: y = yp + yh.
The particular solution is found by proposing a soultion of the same kind as the type of function standing to the right of the equal sign.

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## Cylindrical coordinates

Cylindrical coordinates can be considered as a hybrid between spherical coordinates and rectangular coordinates.

The coordinates of a point is given by the angle between the projection of the point in the xy-plame Φ and two distances: the distance from the point to the xy-plane (z) and the distance to the z-axis (ρ).

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## Dimensions and different coordinate sytems

The number of figures necessary for specifying the position of a point is called the dimension of the space.

For a two dimensional space it is sufficient to use two numbers (x, y) : one specifying the position on a right ot left scale and the other number giving the position on the up- and down scale. (x,y) are often referred to as the cartesian coordinates of a point.

Alternatively one can use polar coordinates where one uses the angle between an arrow pointing at the point and the length of the arrow (r, φ).

These are related by x=r cos(φ) and y =rsin(φ)

In three-dimensional space, our world for example, consequently three numbers are necessary. (x, y, z)

An alternative is the spherical-polar coordinates involving the radius, the azimuthal angle (φ) and the the declination (θ).

These coordinates can be found from the cartesian coordinates by using the formulas
(Pythagorean theorem in three dimensions)