What else Mathematica can do?:
Integral Equation Solver
Ummu Tasnim Husin
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How to do simple integration
To compute the indefinite integral
Integrate[(t^2)Cos[t],t]
To compute the definite integral
Integrate[(t^2)Cos[t],{t,0,Pi}]
Problem: Some definite integration has no closed form
Integrate[x^x,{x,0,1}]
Solution: numerical approximation from series
N[Integrate[x^x,{x,0,1}]]
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Classifications of Integral Equations
An integral equation is an equation in which the unknown function
sign
■ Fredholm integral equation of the first kind (FIE1)
■ Fredholm integral equation of the second kind (FIE2)
■ Volterra integral equation of the first kind (VIE1)
■ Volterra integral equation of the first kind (VIE2)
If
the integral equations are homogeneous.
appears under the integral
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Numerical quarature (integral) formulas
A very simple example for quadrature rule
■ The x[i] points are equally spaced
These quadrature rules are the methods for approximating an integration:
1. Midpoint rule
2. Trapezoidal rule
3. Simpson’s rule
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Midpoint rule
where
This is the illustration of midpoint rule
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Codes: Midpoint rule
$Line = 1;
n=2;
a=0;
b=Pi;
h=N[(b-a)/n];
k[t_]=(t^2)Cos[t];
Do[xmp[i]=N[h (i-1/2)],{i,1,n}];
Sum[h k[xmp[i]],{i,1,n}]
N[-2 Pi]
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Trapezoidal rule
where
This is the illustration of rectangle rule
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Codes: Trapezoidal rule
$Line = 1;
n=2;
a=0;
b=Pi;
h=N[(b-a)/n];
k[t_]=(t^2)Cos[t];
Do[xt[i]=N[h i],{i,0,n}]
d[0]=1; d[n]=1;
Do[d[i]=2,{i,1,n-1}]
Sum[(h/2) d[i]k[xt[i]],{i,0,n}]
N[-2 Pi]
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Simpson’s rule
where
This is the illustration of Simpson' s rule
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Codes: Simpson’s rule
$Line = 1;
n=2;
a=0;
b=Pi;
hh=N[(b-a)/(2n)];
k[t_]:=(t^2) Cos[t];
xs[0]=a;
Do[xs[2i-1]=N[hh (2i-1)];
xs[2i]=N[hh (2i)],{i,1,n}]
ds[0]=1;
ds[2n]=1;
Do[ds[2i-1]=4,{i,1,n}]
Do[ds[2 i]=2,{i,1,n-1}]
Sum[(hh/3) ds[i]k[xs[i]],{i,0,2n}]
N[-2 Pi]
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■ The x[i] points are not equally spaced
■ Gaussian Quadrature
■
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Integrate[t Cos[t],{t,0,Pi}]
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Fun exercise: Evaluate
using Gaussian quadrature rule
(STAY TUNED FOR HINT)
<< NumericalDifferentialEquationAnalysis`
$Line=1;
n=2;
a=0;
b=1;
G=GaussianQuadratureWeights[n,a,b]
{{0.211325, 0.5}, {0.788675, 0.5}}
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Nystrom method
which leads to
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Discretization using
Trapezoidal rule
where
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Mathematica codes:Nystrom method
with Trapezoidal rule
$Line = 1;
n=2;
a=0;
b=1;
h=N[(b-a)/n];
k[x_,t_]:=1-x Cos[x t];
g[x_]:=Sin[x];
Do[x[i]=N[h i]; gg[i]=N[g[x[i]]],{i,0,n}]
b=Table[gg[i],{i,0,n}]
b//MatrixForm
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For[i=0,i≤n,++i,
For[j=0,j≤n,++j,
K[i,j]=N[k[x[i],x[j]]]
]
]
KN=Table[K[i,j],{i,0,n},{j,0,n}];
KN//MatrixForm
d[0]=1; d[n]=1;
Do[d[i]=2,{i,1,n-1}]
DM=N[(h/2)]DiagonalMatrix[Table[d[i],{i,0,n}]];
DM//MatrixForm
KN.DM//MatrixForm
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ID=IdentityMatrix[n+1];
ID//MatrixForm
A=ID-KN.DM;
A//MatrixForm
f=N[LinearSolve[A,b]];
f//MatrixForm
Do[xs[i]=N[1],{i,0,n}]
Max[Table[Abs[f[[i+1]]-xs[i]],{i,0,n}]]
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Mathematica codes:Nystrom method
with Simpson’s rule
$Line=1;
n=2;
a=0;
b=1;
h=N[(b-a)/(2n)];
Do[x[i]=N[h i],{i,0,2n}]
k[x_,t_]:=1-x Cos[x t];
For[i=0,i≤2n,++i,
For[j=0,j≤2n,++j,
kk[i,j]=N[k[x[i],x[j]]]
]
]
MK=Table[kk[i,j],{i,0,2n},{j,0,2n}];
MK//MatrixForm
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ID=IdentityMatrix[2n+1];
ID//MatrixForm
d[0]=1;d[2n]=1;
Do[d[2i]=2,{i,1,n-1}]
Do[d[2i-1]=4,{i,1,n}]
MD=N[(h/3)DiagonalMatrix[Table[d[i],{i,0,2n}]]];
MD//MatrixForm
g[x_]:=Sin[x];
Do[ gg[i]=N[g[x[i]]],{i,0,2n}]
VG=Table[gg[i],{i,0,2n}];
VG//MatrixForm
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MA=ID-MK.MD;
MA//MatrixForm
VF=N[LinearSolve[MA,VG]];
VF//MatrixForm
Do[ef[i]=N[1],{i,0,2n}]
EF=Table[ef[i],{i,0,2n}];
EF//MatrixForm;
Max[Table[Abs[VF[[i+1]]-EF[[i+1]]],{i,0,2n}]]
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Mathematica codes:Nystrom method
with Gaussian rule
<<NumericalDifferentialEquationAnalysis`
$Line=1;
n=2;
a=0;
b=1;
G=GaussianQuadratureWeights[n,a,b]
Do[x[i]=G[[i,1]];w[i]=G[[i,2]],{i,1,n}]
ID=IdentityMatrix[n];
ID//MatrixForm
k[x_,t_]:=1-x Cos[-x t];
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For[i=1,i≤n,++i,
For[j=1,j≤n,++j,
kk[i,j]=N[k[x[i],x[j]]]
]
]
MK=Table[kk[i,j],{i,1,n},{j,1,n}];
MK//MatrixForm
Do[d[i]=w[i],{i,1,n}]
MD=DiagonalMatrix[Table[d[i],{i,1,n}]];
MD//MatrixForm
g[x_]:=Sin[x];
Do[ gg[i]=N[g[x[i]]],{i,1,n}]
VG=Table[gg[i],{i,1,n}];
VG//MatrixForm
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MA=ID-MK.MD;
MA//MatrixForm
VF=N[LinearSolve[MA,VG]];
VF//MatrixForm
Do[ef[i]=N[1],{i,1,n}]
EF=Table[ef[i],{i,1,n}];
EF//MatrixForm
Max[Table[Abs[VF[[i]]-EF[[i]]],{i,1,n}]]
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On the method of solving interior Riemann-Hilbert problem
on simply connected region
Assumming the solution in the following form
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the problem is now reduced into the form of integral equation
where
$Line=1;
n=512;
Do[t[i]=N[2Pi (i-1)/n];
A[i]=N[Exp[-I t[i]]];
DA[i]=N[-I A[i]];
R[i]=N[3+Cos[2 t[i]]+Sin[5 t[i]]];
DR[i]=N[-2 Sin[2 t[i]]+5 Cos[5 t[i]]];
DDR[i]=N[-4 Cos[2 t[i]]-25 Sin[5 t[i]]];
e[i]=N[R[i]Exp[I t[i]]];
De[i]=N[I e[i]+DR[i]Exp[I t[i]]];
DDe[i]=N[I De[i]+DR[i]I Exp[I t[i]]+DDR[i]Exp[I t[i]]];
g[i]=4 R[i],{i,1,n}]
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VG=Table[g[i],{i,1,n}];
VG//MatrixForm;
For[i=1,i≤n,++i,
For[j=1,j≤n,++j,
If[i⩵j,mI[i,i]=1,mI[i,j]=0
]
]
]
MI = Table[mI[i,j],{i,1,n},{j,1,n}];
MI//MatrixForm;
For[i=1,i≤n,++i,
For[j=1,j≤n,++j,
mN[i,j]=N[(2/n)Im[(A[i]/A[j])De[j]/(e[j]-e[i])]]
]
]
MN = Table[mN[i,j],{i,1,n},{j,1,n}];
MN//MatrixForm;
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MM = MI + MN;
MM//MatrixForm;
VGT=N[LinearSolve[MM,VG]];
VGT//MatrixForm;
PHIe[u_]=(2)u;
PHInA[w_]= Sum[De[i]VGT[[i]]/(A[i](e[i]-w)),{i,1,n}]/Sum[De[i]/(e[i]-w)
Error=N[Abs[PHIe[-1.66-1.57I]-PHInA[-1.66-1.57I]]]
Error=N[Abs[PHIe[1.12-0.23I]-PHInA[1.12-0.23I]]]
Error=N[Abs[PHIe[3.94+0.81I]-PHInA[3.94+0.81I]]]
Error=N[Abs[PHIe[-0.99+0.99I]-PHInA[-0.99+0.99I]]]
Error=N[Abs[PHIe[0.1-I]-PHInA[0.1-I]]]
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PHINN[zz_]:=PHInA[zz]
PHIN[x_,y_]:=PHINN[x+I y]
PHIEE[zz_]:=PHIe[zz]
PHIE[x_,y_]:=PHIEE[x+I y]
Plot3D[{N[Abs[PHIE[x,y]-PHIN[x,y]]],0},{x,-5,5},{y,-3,3},
RegionFunction→Function[{x,y,z},
Sqrt[x^2+y^2]<3+Cos[2 ArcTan[x,y]]+Sin[5 ArcTan[x,y]]],
AxesLabel→{Re t,Im t,Null},LabelStyle→Large,BoxRatios→{6,4,1.5},PlotRange
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Appreciation goes to
- ICMMM2016 organizer
- My supervisors
- My research group members
[email protected]
Billik Pasca-Siswazah, Level 4, C17
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Thank you for your kind
attention