An exact solution method for Fredholm integro-differential equations
Author
Kyriaki D Tsilika
Title
An exact solution method for Fredholm integro-differential equations
Description
A solver for unique solutions of Fredholm integro-differential equations that displays analytical formulations that can be called up directly.
Category
Academic Articles & Supplements
Keywords
boundary value problems, ordinary differential equations, Fredholm integro-differential equations, multipoint boundary conditions, nonlocal integral boundary conditions, correct operators, exact solutions
URL
http://www.notebookarchive.org/2019-10-38g3liq/
DOI
https://notebookarchive.org/2019-10-38g3liq
Date Added
2019-10-07
Date Last Modified
2019-10-07
File Size
100.85 kilobytes
Supplements
Rights
CC BY-NC-SA 4.0
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This notebook computes the unique solution of the integro-differential equations of the examples 1 and 2 of the paper
An exact solution method for Fredholm integro - differential equations
An exact solution method for Fredholm integro - differential equations
We develop a solver for unique solutions of Fredholm integro-differential equations in a Mathematica notebook that displays analytical formulations that can be called up directly. Our easy-to-use program provides in one entry, exact solutions for the abstract operator equation
Bu =u-gF(u) = f, D(B) = {u ϵ D() : Φ(u) = NΨ(u)g}, u ϵ D(), f ϵ Y, where X, Y be complex Banach spaces, A: XY is an ordinary m-order differential operator with finite dimensional kernel z=(, … ) which is a basis of kerA and the components of the functional vectors Φ=col(, …) ϵ, Ψ=col(, …) ϵ, F=col(, …) ϵ, a vector g=(, … )ϵ and , … is a linearly independent set, N is a m×l matrix.
Bu =
A
A
A
A
z
1
z
m
Φ
1
Φ
m
*
[]
m
X
Ψ
1
Ψ
l
*
[]
m
X
F
1
F
n
n
[]
*
Y
g
1
g
n
n
Y
g
1
g
n
The cells with the green background are used to assign values to problem’s parameters and to define the structural elements of the operator equation. Unformatted cells contain necessary input for the results of the study. The output consists of the following results:
1. W,V=the matrices in the condition for the injectivity of B (existence condition also)
2. Det[V], Det[W]=Determinants of W,V needed for the necessary and sufficient condition of injectivity of B
3. Automated testing for injectivity of B or the existence criterion
4. solution= the analytic solution of Fredholm integro-differential equation
5. Plot the solution over the region of the initial conditions
The relevant output is created in a way as to be interpreted without the knowledge of the theoretical methodology. The criterion for injectivity of B that is tested and verified is the only requiremen to apply Theorem 2 of [1] and formulate the unique solution.
1. W,V=the matrices in the condition for the injectivity of B (existence condition also)
2. Det[V], Det[W]=Determinants of W,V needed for the necessary and sufficient condition of injectivity of B
3. Automated testing for injectivity of B or the existence criterion
4. solution= the analytic solution of Fredholm integro-differential equation
5. Plot the solution over the region of the initial conditions
The relevant output is created in a way as to be interpreted without the knowledge of the theoretical methodology. The criterion for injectivity of B that is tested and verified is the only requiremen to apply Theorem 2 of [1] and formulate the unique solution.
Example 1
Example 1
In[]:=
(*GivetheordermofthedifferentialoperatorA/numberofconditionsorthedimensionmofthefunctionalvectorΦ*)
In[]:=
m=2;
In[]:=
(*GivethenumberoffunctionsorthedimensionlofthefunctionalvectorΨ*)
In[]:=
l=2;
In[]:=
(*GivethedimensionnofthefunctionalvectorF*)
In[]:=
n=1;
In[]:=
(*DefinethestructuralelementsoftheoperatorequationAu-gF(Au)=f*)
In[]:=
(*FistheintegralpartoftheIDE*)
In[]:=
F[function_]:=x*functionx
1
∫
0
In[]:=
f[t_]:={-6*t^3+4*t+4}
In[]:=
g[t_]:={3*t^3-2*t}
In[]:=
(*Givethevaluesofthevariablesintheboundaryconditions*)
In[]:=
ti:={1,1}
In[]:=
(*Givethem×lNmatrixnmatrixsuchthatΦ(u)=NΨ(u)*)
In[]:=
nmatrix={{2,-3},{-1,5}}
Out[]=
{{2,-3},{-1,5}}
In[]:=
(*GivethefunctionalvectorΨsuchthatΦ(u)=NΨ(u)*)
In[]:=
Ψ[function_]:={function/.t->ti[[1]],D[function,t]/.t->ti[[2]]}
In[]:=
(*Thesolutionprocedure*)
In[]:=
W:=IdentityMatrix[n]-F[g[x]]
In[]:=
V:=IdentityMatrix[l]-Ψ[z].nmatrix
In[]:=
z:=Table[t^i/i!,{i,0,m-1}]
In[]:=
inverseA[function_]:=*functionx
1
(m-1)!
t
∫
0
m-1
(t-x)
(*Verifytheassumptionsoftheorem2*)
In[]:=
(*TestingnecessaryandsufficientconditionsforoperatorBu=Au-gF(Au)tobeinjective*)
In[]:=
Det[W]
Out[]=
16
15
In[]:=
Det[V]
Out[]=
2
In[]:=
(*Testingtheexistenceanduniquenesscriteria*)
In[]:=
If[Det[W]≠0&&Det[V]≠0,"The IDE has a unique solution","The solution is not unique"]
Out[]=
The IDE has a unique solution
In[]:=
(*Hereistheuniquesolutionbytheexactsolutionmethod*)
In[]:=
solution:=Simplify[inverseA[f[x]]+(inverseA[g[x]]+z.nmatrix.Inverse[V].Ψ[inverseA[g[x]]]).Inverse[W].F[f[x]]+z.nmatrix.Inverse[V].Ψ[inverseA[f[x]]]]
In[]:=
Print["The exact solution of the IDE is"Flatten[solution]]
{The exact solution of the IDE is(3-5t+2)}
2
t
In[]:=
Expand[solution[[1]]]
Out[]=
3-5t+2
2
t
In[]:=
Plot[%,{t,0,1},AxesLabel{t,solution[[1]]},PlotLabel"u(t) over the region of the initial conditions"]
Out[]=
Example 2
Example 2
In[]:=
(*GivetheordermofthedifferentialoperatorA/numberofconditionsorthedimensionmofthefunctionalvectorΦ*)
In[]:=
m=4;
In[]:=
(*GivethenumberoffunctionsorthedimensionlofthefunctionalvectorΨ*)
In[]:=
l=4;
In[]:=
(*GivethedimensionnofthefunctionalvectorF*)
In[]:=
n=1;
In[]:=
(*DefinethestructuralelementsoftheoperatorequationAu-gF(Au)=f*)
In[]:=
(*FistheintegralpartoftheIDE*)
In[]:=
F[function_]:=x*functionx
1
∫
0
In[]:=
f[t_]:={-12*t^2+48*t+12}
In[]:=
g[t_]:={t^2-4*t+1}
In[]:=
(*Givethevaluesofthevariablesintheboundaryconditions*)
In[]:=
ti={1,1,1,1};
In[]:=
(*Givethem×lNmatrixnmatrixsuchthatΦ(u)=NΨ(u)*)
In[]:=
nmatrix={3,0,0,0},{0,-5,1,0},0,0,-,0,{0,0,24,-10}
1
5
Out[]=
{3,0,0,0},{0,-5,1,0},0,0,-,0,{0,0,24,-10}
1
5
In[]:=
(*GivethefunctionalvectorΨsuchthatΦ(u)=NΨ(u)*)
In[]:=
Ψ[function_]:={function/.t->ti[[1]],D[function,t]/.t->ti[[2]],D[function,{t,2}]/.tti[[3]],D[function,{t,3}]/.tti[[4]]}
In[]:=
(*Thesolutionmethod*)
In[]:=
W:=IdentityMatrix[n]-F[g[x]]
In[]:=
V:=IdentityMatrix[l]-Ψ[z].nmatrix
In[]:=
z:=Table[t^i/i!,{i,0,m-1}]
In[]:=
inverseA[function_]:=*functionx
1
(m-1)!
t
∫
0
m-1
(t-x)
In[]:=
(*TestingnecessaryandsufficientconditionsforoperatorBu=Au-gF(Au)tobeinjective*)
In[]:=
Det[W]
Out[]=
19
12
In[]:=
Det[V]
Out[]=
648
5
In[]:=
(*Testingtheexistenceanduniquenesscriterion*)
In[]:=
If[Det[W]≠0&&Det[V]≠0,"The IDE has a unique solution","The solution is not unique"]
Out[]=
The IDE has a unique solution
In[]:=
(*Hereistheuniquesolutionbytheexactsolutionmethod*)
In[]:=
solution:=Simplify[inverseA[f[x]]+(inverseA[g[x]]+z.nmatrix.Inverse[V].Ψ[inverseA[g[x]]]).Inverse[W].F[f[x]]+z.nmatrix.Inverse[V].Ψ[inverseA[f[x]]]]
In[]:=
Print["The exact solution of the IDE is"Flatten[solution]]
{The exact solution of the IDE is(-1+)}
2
t
2
t
In[]:=
Expand[solution[[1]]]
Out[]=
-+
2
t
4
t
In[]:=
Plot[%,{t,0,1},AxesLabel{"t",solution[[1]]},PlotLabel"u(t) over the region of the initial conditions"]
Out[]=
References
References
1. Tsilika K.D. 2019. An exact solution method for Fredholm integro - differential equations. Information and Control Systems 4 : 2–8. DOI : 10.31799/1684 - 8853 - 2019 - 4 - 2 - 8.
Cite this as: Kyriaki D Tsilika, "An exact solution method for Fredholm integro-differential equations" from the Notebook Archive (2019), https://notebookarchive.org/2019-10-38g3liq
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