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APPLICATION OF TAYLOR’S SERIES
AND PARTIAL DIFFRENTIAL
EQUATION
PRESENTED BY—
DHEERENDRA RAJPUT
ABSTRACT-----
• Polynomial functions are easy to understand but complicated
functions, infinite polynomials, are not obvious. Infinite
polynomials are made easier when represented using series:
complicated functions are easily represented using Taylor’s series.
This representation make some functions properties easy to study
such as the asymptotic behavior. Differential equations are made
easy with Taylor series. Taylor’s series is an essential theoretical
tool in computational science and approximation. This paper
points out and attempts to illustrate some of the many
applications of Taylor’s series expansion. Concrete examples in the
physical science division and various engineering fields are used to
paint the applications pointed out.
Introduction of Taylor Series
. Taylors series is an expansion of a function into an
infinite series of a variable x or into a finite series plus
a remainder term[1]. The coefficients of the
expansion or of the subsequent terms of the series
involve the successive derivatives of the function. The
function to be expanded should have a nth derivative
in the interval of expansion. The series resulting from
Taylors expansion is referred to as the Taylor series.
The series is finite and the only concern is the
magnitude of the remainder. :
EVALUATING DEFINITE
INTEGRALS
Some functions have no anti-derivative which can be
expressed in terms of familiar functions. This makes
evaluating definite integrals of these functions difficult
because the fundamental theorem of calculus cannot
be
used.
Example=
Taylor series
sin(x^2) = x^2 – x^6/3! + x^10/5! ………
(MATLAB)
The Taylor series can then be integrated:
0
1
sin 𝑥2
𝑑𝑥 =
𝑥3
3
+
𝑥7
7𝑋3!
+
𝑥11
11𝑋5!
+ ⋯ … .
(MATLAB)
syms x
S(x) = sin(x.^2)
T = taylor(S,x,'Order',18)
I = int(T,x)
MATLAB syms x
S(x) = sin(x.^2)
T = taylor(S,x,'Order',18)
• This is an alternating series and by adding all the terms the series
converges to 0.31026
MATLAB CODE—
syms x
S(x) = sin(x.^2)
T = taylor(S,x,'Order',18)
I = int(T,x,[0 ])
UNDERSTANDING ASYMPTOTIC
BEHAVIOR
The electric field
obeys the inverse square law.
E =Kq/r^2
Where E is the electric field, q is the charge, r is the
distance away from the charge and k is some constant
of proportionality. Two opposite charges placed side by
side, setup an electric dipole moment
Taylor’s series is used to study this behavior.
• E = Kq/(x-r)^2 + Kq/(x+r)^2
• An electric field further away from the dipole is obtained after expanding
the terms in the denominator—
• E = Kq/x^2(1-r/x)^2 – Kq/x^2(1+r/x)^2
• Taylor’s series can be used to expand the denominators if(x>>>r)
Kq/(1-r/x)^2 = (MATLAB)—
syms x r
E(x) = 1/(1-r/x)^2
T = taylor(E,r,'Order',5)
Kq/(1+r/x)^2 = (MATLAB)
syms x r
E(x) = 1/(1+r/x)^2
T = taylor(E,r,'Order',5--)
EXAMPLES OF APPLICATIONS OF TAYLOR
SERIES--
1----
• The Gassmann relations of poroelasticity provide a connection between
the dry and the saturated elastic moduli of porous rock and are useful in a
variety of petroleum geoscience applications . Because some uncertainty is
usually associated with the input parameters, the propagation of error in
the inputs into the final moduli estimates is immediately of interest. Two
common approaches to error propagation include: a first-order Taylor’s
series expansion and Monte-Carlo methods. The Taylor’s series approach
requires derivatives, which are obtained either analytically or numerically
and is usually limited to a first-order analysis. The formulae for analytical
derivatives were often prohibitively complicated before modern symbolic
computation packages became prevalent but they are now more
accessible.
2----
A numerical method for simulations of nonlinear surface water waves over variable
bathymetry (study of underwater depth of third dimension of lake or ocean floor) and which
is applicable to either two- or three dimensional flows, as well as to either static or moving
bottom topography, is based on the reduction of the problem to a lower-dimensional
Hamiltonian system involving boundary quantities alone. A key component of this
formulation is the Dirichlet-Neumann operator (used in analysing boundary conditions e.g
fluid dynamics and crystal growth) which, in light of its joint analyticity properties with
respect to surface and bottom deformations, is computed using its Taylor’s series
representation. The new stabilized forms for the Taylor terms, are efficiently computed by a
pseudo spectral method using the fast Fourier transform
Solving Partial Differential Equations--
In a partial differential equation (PDE), the function
being solved for depends on several variables, and
the differential equation can include partial
derivatives taken with respect to each of the
variables. Partial differential equations are useful for
modelling waves, heat flow, fluid dispersion, and
other phenomena with spatial behavior that changes
over time.
Example: The Heat Equation
• An example of a parabolic PDE is the heat equation in one dimension:
• ∂u/∂t=∂^2u/∂x^2
• Solve heat equation in MATLAB
•solutionProcess
To solve PDEs with pdepe, you must define the equation coefficients for c, f, and s, the
initial conditions, the behavior of the solution at the boundaries, and a mesh of points to
evaluate the solution on
Local Functions
function [c,f,s] = heatpde(x,t,u,dudx)
c = 1; f = dudx; s = 0;
End
function u0 = heatic(x)
u0 = 0.5; end
function
[pl,ql,pr,qr] = heatbc(xl,ul,xr,ur,t)
pl = ul;
ql = 0;
pr = ur - 1;
qr = 0;
end
Select Solution Mesh
Use a spatial mesh of 20 points and a time mesh of 30 points. Since the solution
rapidly
reaches a steady state, the time points near t=0 are more closely spaced together to
capture
this behavior in the output.
L = 1; x = linspace(0,L,20); t = [linspace(0,0.05,20), linspace(0.5,5,10)];
Solve Equation
Finally, solve the equation using the symmetry m, the PDE equation, the initial condition, the boundary
conditions, and the meshes for x and t.
m = 0;
sol = pdepe(m,@heatpde,@heatic,@heatbc,x,t);
Plot Solution
Use imagesc to visualize the solution matrix.
colormap hot
imagesc(x,t,sol)
colorbar
xlabel('Distance x','interpreter','latex')
ylabel('Time t','interpreter','latex')
title('Heat Equation for $0 le x le 1$ and $0 le t le 5$','interpreter','latex')
CONCLUSION.
Probably the most important application of Taylor series
is to use their partial sums to approximate functions.
These partial sums are (finite) polynomials and are easy
to compute.
As far as I know, the concept of Taylor series was
discovered by the Scottish mathematician James
Gregory and formally introduced by the English
mathematician Brook Taylor in 1715. However, my main
curiosity is about the problems and situations that
resulted in a need to approximate a function using Taylor
series.
•THANKU YOU

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Maths ppt partial diffrentian eqn

  • 1. APPLICATION OF TAYLOR’S SERIES AND PARTIAL DIFFRENTIAL EQUATION PRESENTED BY— DHEERENDRA RAJPUT
  • 2. ABSTRACT----- • Polynomial functions are easy to understand but complicated functions, infinite polynomials, are not obvious. Infinite polynomials are made easier when represented using series: complicated functions are easily represented using Taylor’s series. This representation make some functions properties easy to study such as the asymptotic behavior. Differential equations are made easy with Taylor series. Taylor’s series is an essential theoretical tool in computational science and approximation. This paper points out and attempts to illustrate some of the many applications of Taylor’s series expansion. Concrete examples in the physical science division and various engineering fields are used to paint the applications pointed out.
  • 3. Introduction of Taylor Series . Taylors series is an expansion of a function into an infinite series of a variable x or into a finite series plus a remainder term[1]. The coefficients of the expansion or of the subsequent terms of the series involve the successive derivatives of the function. The function to be expanded should have a nth derivative in the interval of expansion. The series resulting from Taylors expansion is referred to as the Taylor series. The series is finite and the only concern is the magnitude of the remainder. :
  • 4. EVALUATING DEFINITE INTEGRALS Some functions have no anti-derivative which can be expressed in terms of familiar functions. This makes evaluating definite integrals of these functions difficult because the fundamental theorem of calculus cannot be used.
  • 5. Example= Taylor series sin(x^2) = x^2 – x^6/3! + x^10/5! ……… (MATLAB) The Taylor series can then be integrated: 0 1 sin 𝑥2 𝑑𝑥 = 𝑥3 3 + 𝑥7 7𝑋3! + 𝑥11 11𝑋5! + ⋯ … . (MATLAB) syms x S(x) = sin(x.^2) T = taylor(S,x,'Order',18) I = int(T,x) MATLAB syms x S(x) = sin(x.^2) T = taylor(S,x,'Order',18)
  • 6. • This is an alternating series and by adding all the terms the series converges to 0.31026 MATLAB CODE— syms x S(x) = sin(x.^2) T = taylor(S,x,'Order',18) I = int(T,x,[0 ])
  • 7. UNDERSTANDING ASYMPTOTIC BEHAVIOR The electric field obeys the inverse square law. E =Kq/r^2 Where E is the electric field, q is the charge, r is the distance away from the charge and k is some constant of proportionality. Two opposite charges placed side by side, setup an electric dipole moment
  • 8. Taylor’s series is used to study this behavior. • E = Kq/(x-r)^2 + Kq/(x+r)^2 • An electric field further away from the dipole is obtained after expanding the terms in the denominator— • E = Kq/x^2(1-r/x)^2 – Kq/x^2(1+r/x)^2 • Taylor’s series can be used to expand the denominators if(x>>>r) Kq/(1-r/x)^2 = (MATLAB)— syms x r E(x) = 1/(1-r/x)^2 T = taylor(E,r,'Order',5) Kq/(1+r/x)^2 = (MATLAB) syms x r E(x) = 1/(1+r/x)^2 T = taylor(E,r,'Order',5--)
  • 9. EXAMPLES OF APPLICATIONS OF TAYLOR SERIES-- 1---- • The Gassmann relations of poroelasticity provide a connection between the dry and the saturated elastic moduli of porous rock and are useful in a variety of petroleum geoscience applications . Because some uncertainty is usually associated with the input parameters, the propagation of error in the inputs into the final moduli estimates is immediately of interest. Two common approaches to error propagation include: a first-order Taylor’s series expansion and Monte-Carlo methods. The Taylor’s series approach requires derivatives, which are obtained either analytically or numerically and is usually limited to a first-order analysis. The formulae for analytical derivatives were often prohibitively complicated before modern symbolic computation packages became prevalent but they are now more accessible.
  • 10. 2---- A numerical method for simulations of nonlinear surface water waves over variable bathymetry (study of underwater depth of third dimension of lake or ocean floor) and which is applicable to either two- or three dimensional flows, as well as to either static or moving bottom topography, is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving boundary quantities alone. A key component of this formulation is the Dirichlet-Neumann operator (used in analysing boundary conditions e.g fluid dynamics and crystal growth) which, in light of its joint analyticity properties with respect to surface and bottom deformations, is computed using its Taylor’s series representation. The new stabilized forms for the Taylor terms, are efficiently computed by a pseudo spectral method using the fast Fourier transform
  • 11. Solving Partial Differential Equations-- In a partial differential equation (PDE), the function being solved for depends on several variables, and the differential equation can include partial derivatives taken with respect to each of the variables. Partial differential equations are useful for modelling waves, heat flow, fluid dispersion, and other phenomena with spatial behavior that changes over time.
  • 12. Example: The Heat Equation • An example of a parabolic PDE is the heat equation in one dimension: • ∂u/∂t=∂^2u/∂x^2 • Solve heat equation in MATLAB •solutionProcess To solve PDEs with pdepe, you must define the equation coefficients for c, f, and s, the initial conditions, the behavior of the solution at the boundaries, and a mesh of points to evaluate the solution on
  • 13. Local Functions function [c,f,s] = heatpde(x,t,u,dudx) c = 1; f = dudx; s = 0; End function u0 = heatic(x) u0 = 0.5; end function [pl,ql,pr,qr] = heatbc(xl,ul,xr,ur,t) pl = ul; ql = 0; pr = ur - 1; qr = 0; end
  • 14. Select Solution Mesh Use a spatial mesh of 20 points and a time mesh of 30 points. Since the solution rapidly reaches a steady state, the time points near t=0 are more closely spaced together to capture this behavior in the output. L = 1; x = linspace(0,L,20); t = [linspace(0,0.05,20), linspace(0.5,5,10)]; Solve Equation Finally, solve the equation using the symmetry m, the PDE equation, the initial condition, the boundary conditions, and the meshes for x and t. m = 0; sol = pdepe(m,@heatpde,@heatic,@heatbc,x,t); Plot Solution Use imagesc to visualize the solution matrix. colormap hot imagesc(x,t,sol) colorbar xlabel('Distance x','interpreter','latex') ylabel('Time t','interpreter','latex') title('Heat Equation for $0 le x le 1$ and $0 le t le 5$','interpreter','latex')
  • 15.
  • 16. CONCLUSION. Probably the most important application of Taylor series is to use their partial sums to approximate functions. These partial sums are (finite) polynomials and are easy to compute. As far as I know, the concept of Taylor series was discovered by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715. However, my main curiosity is about the problems and situations that resulted in a need to approximate a function using Taylor series.