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few basic differential equation questions ... ??


bimbo36

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i have been following maths like some sort of self learning process ...

 

i was able to improve a lot of things myself ...

 

i was able to understand some basic concepts a little bit more ...

 

i was able to narrow it down like this ...

 

 

 

 

mathematical expressions
equations in one variable
equations in two variables
system of 2 equations containing 2 variables
functions in one variable
functions in two variables

differential equations

first order differential equations
second order differential equations
higher order differential equations ...

linear differential equations
separable differential equations
exact differential equations
homogeneous differential equations
non homogeneous differential equations
using the method of undetermined coefficients ...



partial differential equations ...

 

then i also remember learning a lot of things about linear and non linear equations ...

 

http://www.mathsisfun.com/algebra/systems-linear-equations.html

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

http://www.sosmath.com/soe/SE211105/SE211105.html

 

 

A linear equation is always a polynomial of degree 1 (for example x+2y+3=0). In the two dimensional case, they always form lines; in other dimensions, they might also form planes, points, or hyperplanes. Their "shape" is always perfectly straight, with no curves of any kind. This is why we call them linear equations.

Every other equation is nonlinear. Higher degree polynomials are nonlinear. Trigonometric functions (like sin or cos) are nonlinear. Square roots are nonlinear. The main exception is if the nonlinear piece can evaluate to a constant--for example, sqrt(4)*x is linear because sqrt(4) is just 2, and 2x is linear.

Linear equations have some useful properties, mostly in that they are very easy to manipulate and solve. Although they are quite limited in what they can represent, it is often useful to try and approximate complicated systems using linear equations so that they will be easier to think about and deal with.

Nonlinear equations, for the most part, are much harder to solve and manipulate. Sometimes you need them--nature doesn't always work in straight lines, and nor do mathematicians--but generally speaking, you can only solve nonlinear equations if the systems are fairly small and simple. Solving a linear system with a million interacting variables is very doable with a computer, and most nonlinear solvers aren't going to get even close to that

 

 

 

for example , for a polynomial ...

a solution of a polynomial equation is also called a root of the polynomial ...

a value for the variable that makes the polynomial zero

if you can't find an exact expression, then you can use numerical methods to get approximations ...

with numerical methods you can choose how close to zero you want, and it will give you a value that's at least that close ...

 

 

 

 

 

 

let me add this definition too ...

so that it looks nice in one thread ...

 

 

 

 

The (standard) calculus is broken into two pieces.

i) Differential calculus - which is looking at the instantaneous rates of change of objects with respect to some variables. We have the notion of the derivative of a function.
ii) Integral calculus - which is calculating the area under curves, calculating volumes and so on. This is all given in terms if the (indefinite or definite) integral of a function.

The two notions are tied together via the fundamental theorem of calculus. This says that the derivative and indefinite integral are basically mutual inverses (but not quite).

 

I hope that gives you something to start with.

 

 

 

after all these , i was trying to learn differential equations ...

 

i was able to make some notes ... with the help of some pictures ...

 

can i please keep it , it was fun to read ...

 

differential_equation_33.jpg

differential_equation_examples_33.png

 

 

 

 

 

which category of differential equation does it belong to ??

 

 

first order differential equations
second order differential equations
higher order differential equations ...

linear differential equations
separable differential equations
exact differential equations
homogeneous differential equations
non homogeneous differential equations
using the method of undetermined coefficients ...



partial differential equations ...

 

 

how do i solve this differential equation ??

Edited by bimbo36
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  • 2 weeks later...

let me try to refresh this thread with few more things ...

 

 

i am trying to put all the stuff from my syllabus in this thread , so that i don't have to look anywhere else ...

 

my syllabus was called , "computer oriented numerical methods in c programming language "

 

it had lot of mathematics ...

 

it was a bit of a mess ...

we really lacked proper texts and materials to really understand that subject ....

none of the texts i have read had the proper mathematics or the examples of numerical related programming in c languages properly ...

i was trying to follow these two instead ...

for some understanding of the overall mathematics , before trying to do programming ...

 

http://www.universityofcalicut.info/SDE/BSc_maths_numerical_methods.pdf

 

for programming some maths related things , i would recommend this book ...

Computer Fundamentals and Programming in C - J.B. Dixit

even those books are still lacking many needed information's ...

which is why i am using pictures nowadays to learn it properly ...

let me arrange some things properly in this thread ...

so you too know , how it looked like .. and where it is heading to ...


well it begins with something like this ...

 

 

mathematical expressions
equations in one variable
equations in two variables
system of 2 equations containing 2 variables
functions in one variable
functions in two variables

differential equations

first order differential equations
second order differential equations
higher order differential equations ...

linear differential equations
separable differential equations
exact differential equations
homogeneous differential equations
non homogeneous differential equations
using the method of undetermined coefficients ...



partial differential equations ...

 

http://www.mathsisfun.com/algebra/systems-linear-equations.html

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

http://www.sosmath.com/soe/SE211105/SE211105.html

 

 

A linear equation is always a polynomial of degree 1 (for example x+2y+3=0). In the two dimensional case, they always form lines; in other dimensions, they might also form planes, points, or hyperplanes. Their "shape" is always perfectly straight, with no curves of any kind. This is why we call them linear equations.

Every other equation is nonlinear. Higher degree polynomials are nonlinear. Trigonometric functions (like sin or cos) are nonlinear. Square roots are nonlinear. The main exception is if the nonlinear piece can evaluate to a constant--for example, sqrt(4)*x is linear because sqrt(4) is just 2, and 2x is linear.

Linear equations have some useful properties, mostly in that they are very easy to manipulate and solve. Although they are quite limited in what they can represent, it is often useful to try and approximate complicated systems using linear equations so that they will be easier to think about and deal with.

Nonlinear equations, for the most part, are much harder to solve and manipulate. Sometimes you need them--nature doesn't always work in straight lines, and nor do mathematicians--but generally speaking, you can only solve nonlinear equations if the systems are fairly small and simple. Solving a linear system with a million interacting variables is very doable with a computer, and most nonlinear solvers aren't going to get even close to that

 

 

 

The (standard) calculus is broken into two pieces.

i) Differential calculus - which is looking at the instantaneous rates of change of objects with respect to some variables. We have the notion of the derivative of a function.
ii) Integral calculus - which is calculating the area under curves, calculating volumes and so on. This is all given in terms if the (indefinite or definite) integral of a function.

The two notions are tied together via the fundamental theorem of calculus. This says that the derivative and indefinite integral are basically mutual inverses (but not quite).

 

Differential equations ...

 

 

An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation

 

 

 

Numerical Methods and errors
Interpolation
Numerical Differentiation
Numerical Integration
Solution of Algebraic and Transcendental Equations
Numerical Solution of a system of Linear Equations
Numerical Solution of Ordinary differential equations
Curve fitting
Numerical Solution of problems associated with Partial Differential Equations

 

 

 

Solution of Algebraic and Transcendental Equation
2.1 Introduction
2.2 Bisection Method
2.3 Method of false position
2.4 Iteration method
2.5 Newton-Raphson Method
2.6 Ramanujan's method
2.7 The Secant Method Finite Differences


3.1 Introduction
3.3.1 Forward differences
3.3.2 Backward differences
3.3.3 Central differences
3.3.4 Symbolic relations and separation of symbols
3.5 Differences of a polynomial Interpolation
3.6 Newton's formulae for intrapolation
3.7 Central difference interpolation formulae
3.7.1 Gauss' Central Difference Formulae
3.9 Interpolation with unevenly spaced points
3.9.1 Langrange's interpolation formula
3.10 Divided differences and their properties
3.10.1 Newton's General interpolation formula
3.11 Inverse interpolation Numerical Differentiation and Integration


5.1 Introduction
5.2 Numerical differentiation (using Newton's forward and backward formulae)
5.4 Numerical Integration
5.4.1 Trapizaoidal Rule
5.4.2 Simpson's 1/3-Rule
5.4.3 Simpson's 3/8-Rule Matrices and Linear Systems of equations
6.3 Solution of Linear Systems – Direct Methods
6.3.2 Gauss elimination
6.3.3 Gauss-Jordan Method
6.3.4 Modification of Gauss method to compute the inverse
6.3.6 LU Decomposition
6.3.7 LU Decomposition from Gauss elimination
6.4 Solution of Linear Systems – Iterative methods
6.5 The eigen value problem
6.5.1 Eigen values of Symmetric Tridiazonal matrix Numerical Solutions of Ordinary Differential Equations


7.1 Introduction
7.2 Solution by Taylor's series
7.3 Picard's method of successive approximations
7.4 Euler's method
7.4.2 Modified Euler's Method
7.5 Runge-Kutta method
7.6 Predictor-Corrector Methods
7.6.1 Adams-Moulton Method
7.6.2 Milne's method

 

 

 

for example , for a polynomial ... a solution of a polynomial equation is also called a root of the polynomial ... a value for the variable that makes the polynomial zero if you can't find an exact expression, then you can use numerical methods to get approximations ... with numerical methods you can choose how close to zero you want, and it will give you a value that's at least that close ...

 

 

equations.pngdifferential_equation_answer.pngderivative_integral_differntial_equationbooks.jpg85e7c3b.jpgindex.jpg

 

at this point of time i am not sure how the alphabets , variables , arrays ... etc ...
f(x) ... theta , d^2y/dx^2 , (dy/dx)^3 , integral symbol... are declared ...

again if you use the numerical methods or the appropriate algorithm for the differential equations properly ...

you should be able to find the solution of the problem with a programming language like c ..

 

 

 

my question is , what does numerical method achieve for each one these types of numerical method questions??

 

because the questions ranges from ...

 

polynomials

trigonometric

transcendental

differentiation

integration

differential equations

 

 

i know for a polynomial ...

 

the numerical method means ...

 

 

 

for example , for a polynomial ... a solution of a polynomial equation is also called a root of the polynomial ... a value for the variable that makes the polynomial zero if you can't find an exact expression, then you can use numerical methods to get approximations ... with numerical methods you can choose how close to zero you want, and it will give you a value that's at least that close ...

 

 

 

for differentiation and differential equations ...

 

its about the instantaneous rate of change of the variable x with respect to y

 

then what happens , when you apply numerical methods on their solutions ??

 

what does this approximate values represent ??

 

again is it about the instantaneous rate of change of the variable x with respect to y ...

 

is it always about how close to zero you want the solutions to be ??

 

 

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i am sorry .. this is seriously one last time ... i collected notes like this from many different places and it took a lot of time to arrange it like this ...

 

can i please keep it for one last time ??

 

i wont repeat this ...

 

from next time onward i promise i would keep it short and neat ...

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  • 3 weeks later...

 

 

:) :-) :)

 

 

what are you trying to ask as a question? Can you narrow it done to a simple paragraph without all the packing?

 

 

:) :-) :)

I have almost lost my ways among his posts :) :-) :) :-)

 

 

polynomials

trigonometric

transcendental

differentiation

integration

differential equations

NOTE : The term given in bold is a bit far from your question. and may make you confused

I do not recommend that you use that on the net.

avoid this.

Edited by blue89
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sorry for the long posts ...

 


 

i was able to narrow it down to these ...

 

An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation

they are mainly classified into two ..
ordinary differential equation
partial differential equation ..

then comes first order differential equations to nth order differential equations ...

order is the highest number of the differentiations appearing
degree is the power of the highest order derivative in the equation ...

then there are types of differential equations , depending on their order

separable equations
homogeneous equations
linear equations
exact equations

 

 

A partial differential equation is an equation involving functions and their partial derivatives ...

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant

 


i was very confused when i began studying this ...

 

because i was mostly aiming for the numerical methods ...

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dear bimbo :)

 

really I couldn't find (whether) this last post was containing any question .

 

but I can say theese and give some norifications to you for not to be confused.

 

1) numerical analysis and differential equation course both are very different courses , the differential equation courses contain just the subjects given in your this comment

 

 

 

An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation

they are mainly classified into two ..
ordinary differential equation
partial differential equation ..

then comes first order differential equations to nth order differential equations ...

 

but rhe numerical analsysis course is separate and contains different subjects ,look this course contans subjects like these ;

 

 

--->> calculating computational flaws

--->> calculating computational flaws via using limit forms & derivation

--->> some approximation techniques : like simpsons ,trapez ,interpolation ...etc.

--->> some integral forms & theories

--->> trend

 

NOTE: all these subjects only for cogent for BSc student (not for MSc students or higher..)

 

do not confuse the differential equation course with numerical analysis ,the suitable one for you is of course numerical analysis as I said in my previous post.

 

good results..

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blue89,

 

this thread is a bit of a mess ...

 

i was trying to narrow it down with the help of some pictures ...

 

we had this subject called "computer oriented numerical methods in c programming language " in the second semester ...

 

that was the only maths related subject we had ...

 

the rest was 12 programming languages ...

 

the "computer oriented numerical methods in c programming language " looked a bit like this ...

 

 

Numerical Methods and errors

Interpolation

Numerical Differentiation

Numerical Integration

Solution of Algebraic and Transcendental Equations

Numerical Solution of a system of Linear Equations

Numerical Solution of Ordinary differential equations

Curve fitting

Numerical Solution of problems associated with Partial Differential Equations

 

numerical_methodds.jpg

 

 

 

An Example

Say we were to solve the initial value problem:

y′ = 2x

y(0) = 0

It's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (Yes, I know we don't know how to do this yet, but go with me on this for a second!) The resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.


Using Numerical Solutions

So what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as Mathematica, do do with these solutions. (Mathematica produces a join-the-dots function that it calls InterpolatingFunction.)

When actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: to find the values of the solution at specific points. This is exactly what we can still do with a numerical solution

 

 

numerical_solutions.png

 

euler_algorithm.png

 

 

turev-tanim.jpg

 

ccc_numerical_methods_2.png

 

 

 

 

terms like "Solution of equations by iterative algorithms" were very confusing in the beginning ... now i am a little bit more familiar with all these terms ...

 

 

 

thanks a lot for the replies ...


 

 

 

 

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blue89,

 

1)this thread is a bit of a mess ...

 

we had this subject called "computer oriented numerical methods in c programming language " in the second semester ...

 

2)the "computer oriented numerical methods in c programming language " looked a bit like this ...

 

 

3)thanks a lot for the replies ...

 

 

 

1) :):wub: :wub: :)

 

really.... :wub:

 

although this does not mean any problem for me as a member , I do not know whether you take any notification from editors , but you have not used any bad words in forum,I also think your sentences were a bit lovely even if made me confused :):wub:

 

2) unfortunately I do not know using C programming , but we had been writing such programmes at another programming (Force : Fortran) this probably similar to C.

 

3) look this is lovely expression. :)

 

I personally highly care to politeness :)

 

hımm imatfaal's expression was also pretty style of asking a question :)

Edited by blue89
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