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Taylor series method. C 조 박재무 윤동일 장석민. outline. What is Taylor series method Main Topic 1 Main Topic 2 Future work. Problem 1 Result 1 How to solve problem 1 Summary 1. Problem 2 Result 2 How to solve problem 2 Summary 2. What is Taylor series?.

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taylor series method

Taylorseries method

C조

박재무

윤동일

장석민

outline
outline
  • What is Taylor seriesmethod
  • Main Topic 1
  • Main Topic 2
  • Future work

Problem 1

Result 1

How to solve problem 1

Summary 1

Problem 2

Result 2

How to solve problem 2

Summary 2

slide3

WhatisTaylorseries?

The Taylor series of a real or complex-valued functionƒ(x) that is infinitely differentiable at a real or complex number ais the power series which can be written in the more compact sigma notation as

When a = 0, the series is also called a Maclaurin series

slide4

WhatisTaylorseries?

Taylor seriesis a representation of a function as an infinite sum of terms that are calculated from the values of the function\'s derivative at a single point.

Example

slide5

WhatisTaylorseries method?

Euler’s method

Tangent line local approximation

Taylor series

Higher degree better approximation

graph of e x
Graph of e^x

Taylortool in matlab

Taylor polynomial of degree 1 (Euler’s M)

Taylor polynomial of degree 2

problem 1
Problem 1

Compute the Taylor polynomials of degree 4 for the solutions to the given initial value problems. Use these Taylor polynomials to approximate the solution at x = 1.

result 1
Result 1

Problem(1) has error that can’t be ignored

Problem(2) has large error

subproblem 1 analytical
Subproblem 1 (Analytical)

Integrating factor

how can we decrease the error
How can we decrease the error?

Problem (1)

: Increase the degree of Taylor polynomial

Error : 0.247498 Error : 0.000298

how can we decrease the error1
How can we decrease the error?

Problem (2)

Can not be solve the problem by increasing degree…..

how can we decrease the error2
How can we decrease the error?

Problem (2)

Euler’s method can decrease the error

When mesh size = 0.1,

y(1) = 2.094317

errer = 0.050841

x=0;

y=4;

for i=1:100

yp=y*(2-y);

y1=y+0.1*yp;

y=y1;

x=x+0.01;

end

solve d e in matlab
Solve D.E in matlab

syms x y

f=dsolve(\'Dy=x-2*y\', \'y(0)=1\', \'x\')

p=taylor(f,5,x)

f = x/2 + 5/(4*exp(2*x)) - ¼

p = (5*x^4)/6 - (5*x^3)/3 + (5*x^2)/2 - 2*x + 1

summary 1
Summary 1

Taylor series method can find approximation

But it need higher degree

and some points in some function can not

enable to get approximation

problem 2
Problem 2

Compare the use of Euler’s method with that of Taylor series to approximate the solution (x) to the initial value problem

Do this by completing the following table:

problem 2 euler s method
Problem 2 ( Euler’s method)

x=0;

y=0;

for i=1:10  for i=1:100

slope=-y+cos(x)-sin(x);

y1=y+0.1*slope;  y1=y+0.3*slope;

y=y1;

x=x+0.1;  x=x+0.3;

end

problem 2 taylor s method x1
Problem 2 ( Taylor’s method)(x)

Step size = 1

Step size = 3

Step size = 1

Step size = 3

problem 2 taylor s method1
Problem 2 ( Taylor’s method)

Taylor polynomial

Degree 2

Taylor polynomial

Degree 5

graph degree 2
Graph(degree 2)

y=cos(x)-exp(-x)

y=cos(x)-exp(-x)

Taylor polynomial of degree 2 with step size 0.1

Taylor polynomial of degree 2 with step size 1

problem 2 taylor s method2
Problem 2 ( Taylor’s method)

Degree: 2

Degree: 5

x=0;

y=0;

step=0.1;

for i=1:10

slope=-y+cos(x)-sin(x);

slope2=-2*cos(x)+y;

slope3=cos(x)+sin(x)-y;

slope4=y;

slope5=-y+cos(x)-sin(x);

y1=y+step*slope

+(step)^2/2*slope2

+(step)^3/(3*2*1)*slope3

+(step)^4/(4*3*2*2)*slope4

+(step)^5/(5*4*3*2*1)*slope5;

y=y1;

x=x+step;

end

x=0;

y=0;

step=0.1;

for i=1:10

slope=-y+cos(x)-sin(x);

slope2=-2*cos(x)+y;

y1=y+step*slope+

(step)^2/2*slope2;

y=y1;

x=x+step;

end

problem 2 analytical
Problem 2 ( Analytical)

Integrating factor

summary 2
Summary 2

Good approximation

In same method

- Step size : small

In same step size

- Taylor Method is better than Euler’s Method

- Taylor polynomial of higher degree is better

than Taylor polynomial of lower degree

But Taylor polynomial of higher degree

needs many calculation

think more
Think more….

How can we get approximation

fast and accurate

with simple code(calculation)?

Runge-Kutta

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