A Standard Integral Equation
$begingroup$
Consider the integral equation
$$phi(x) = x + lambdaint_0^1 phi(s),ds$$
Integrating with respect to $x$ from $x=0$ to $x=1$:
$$int_0^1 phi(x),dx = int_0^1x,dx + lambda int_0^1Big[int_0^1phi(s),dsBig],dx$$
which is equivalent to
$$int_0^1 phi(x),dx = frac{1}{2} + lambda int_0^1phi(s),ds$$
How can I go from here in order to solve the problem for the homogeneous case and find the corresponding characteristic values and associated rank?
linear-algebra integration matrix-equations
asked 8 hours ago
LightningStrikeLightningStrike
455
$endgroup$
add a comment |
$begingroup$
Consider the integral equation
$$phi(x) = x + lambdaint_0^1 phi(s),ds$$
Integrating with respect to $x$ from $x=0$ to $x=1$:
$$int_0^1 phi(x),dx = int_0^1x,dx + lambda int_0^1Big[int_0^1phi(s),dsBig],dx$$
which is equivalent to
$$int_0^1 phi(x),dx = frac{1}{2} + lambda int_0^1phi(s),ds$$
How can I go from here in order to solve the problem for the homogeneous case and find the corresponding characteristic values and associated rank?
linear-algebra integration matrix-equations
asked 8 hours ago
LightningStrikeLightningStrike
455
$endgroup$
1
$begingroup$
What is $lambda$? What do you mean by "solve the problem"? I don't see what "the problem" is supposed to mean. Of which object do you want to find the characteristic values and ranks? Have you checked your definition of $phi$?
$endgroup$
– James
7 hours ago
$begingroup$
My apologies, $lambda$ is an arbitrary constant. In essence I want to obtain an expression of $phi(x)$ which does not contain a function of $s$, which the initial integral equation has.
$endgroup$
– LightningStrike
7 hours ago
add a comment |
$begingroup$
Consider the integral equation
$$phi(x) = x + lambdaint_0^1 phi(s),ds$$
Integrating with respect to $x$ from $x=0$ to $x=1$:
$$int_0^1 phi(x),dx = int_0^1x,dx + lambda int_0^1Big[int_0^1phi(s),dsBig],dx$$
which is equivalent to
$$int_0^1 phi(x),dx = frac{1}{2} + lambda int_0^1phi(s),ds$$
How can I go from here in order to solve the problem for the homogeneous case and find the corresponding characteristic values and associated rank?
linear-algebra integration matrix-equations
asked 8 hours ago
LightningStrikeLightningStrike
455
$endgroup$
Consider the integral equation
$$phi(x) = x + lambdaint_0^1 phi(s),ds$$
Integrating with respect to $x$ from $x=0$ to $x=1$:
$$int_0^1 phi(x),dx = int_0^1x,dx + lambda int_0^1Big[int_0^1phi(s),dsBig],dx$$
which is equivalent to
$$int_0^1 phi(x),dx = frac{1}{2} + lambda int_0^1phi(s),ds$$
How can I go from here in order to solve the problem for the homogeneous case and find the corresponding characteristic values and associated rank?
linear-algebra integration matrix-equations
linear-algebra integration matrix-equations
asked 8 hours ago
LightningStrikeLightningStrike
455
asked 8 hours ago
LightningStrikeLightningStrike
455
edited 7 hours ago
LightningStrike
asked 8 hours ago
LightningStrikeLightningStrike
455
asked 8 hours ago
LightningStrikeLightningStrike
455
asked 8 hours ago
LightningStrikeLightningStrike
455
455
1
$begingroup$
What is $lambda$? What do you mean by "solve the problem"? I don't see what "the problem" is supposed to mean. Of which object do you want to find the characteristic values and ranks? Have you checked your definition of $phi$?
$endgroup$
– James
7 hours ago
$begingroup$
My apologies, $lambda$ is an arbitrary constant. In essence I want to obtain an expression of $phi(x)$ which does not contain a function of $s$, which the initial integral equation has.
$endgroup$
– LightningStrike
7 hours ago
add a comment |
1
$begingroup$
What is $lambda$? What do you mean by "solve the problem"? I don't see what "the problem" is supposed to mean. Of which object do you want to find the characteristic values and ranks? Have you checked your definition of $phi$?
$endgroup$
– James
7 hours ago
$begingroup$
My apologies, $lambda$ is an arbitrary constant. In essence I want to obtain an expression of $phi(x)$ which does not contain a function of $s$, which the initial integral equation has.
$endgroup$
– LightningStrike
7 hours ago
1
1
$begingroup$
What is $lambda$? What do you mean by "solve the problem"? I don't see what "the problem" is supposed to mean. Of which object do you want to find the characteristic values and ranks? Have you checked your definition of $phi$?
$endgroup$
– James
7 hours ago
$begingroup$
What is $lambda$? What do you mean by "solve the problem"? I don't see what "the problem" is supposed to mean. Of which object do you want to find the characteristic values and ranks? Have you checked your definition of $phi$?
$endgroup$
– James
7 hours ago
$begingroup$
My apologies, $lambda$ is an arbitrary constant. In essence I want to obtain an expression of $phi(x)$ which does not contain a function of $s$, which the initial integral equation has.
$endgroup$
– LightningStrike
7 hours ago
$begingroup$
My apologies, $lambda$ is an arbitrary constant. In essence I want to obtain an expression of $phi(x)$ which does not contain a function of $s$, which the initial integral equation has.
$endgroup$
– LightningStrike
7 hours ago
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
Relabelling the dummy variable $xmapsto s$ on the LHS of your final equation, $$int_0^1phi(s),ds-lambdaint_0^1phi(s),ds=frac12\implies int_0^1phi(s),ds=frac1{2(1-lambda)}$$
Thus $$phi(x)=x+fraclambda{2(1-lambda)}$$
answered 7 hours ago
John DoeJohn Doe
11.3k11239
$endgroup$
add a comment |
$begingroup$
Note $int_{0}^{1}{phi(s)ds}$ is a constant say $a$. Your functional equation (FE) can be rewritten as: $$phi(x)=x+alambda$$
Putting into FE yields:
$$x+alambda=x+lambdaint_{0}^{1}{(s+alambda )ds }iff alambda=lambdabig(frac{1}{2}+lambda abig)$$
If $lambda=0$ then $phi(x)=x$
if $lambdane 1$ $a=frac{1}{2}+lambda aiff ( 1-lambda)a=frac{1}{2}iff a=frac{1}{2-2lambda}$ and then $phi(x)=x+frac{lambda}{2-2lambda}$
If $lambda=1$ there won’t besuch $phi$.
answered 7 hours ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
1,478211
$endgroup$
add a comment |
$begingroup$
If you are after finding $phi(x)$, one approach that comes to mind is to assume it is smooth enough to have a normally convergent (so we can interchange series summation and integration) Taylor expansion on $[0, 1]$:
$$
phi(x) = sum_{n geq 0} a_{n} x^{n}.
$$
Substituting it into your equation, we get:
$$
sum_{n geq 0} a_{n} x^{n} = x + lambda sum_{n geq 0}{a_{n} over n+1}.
$$
Matching up the coefficients of the difference powers of $x$, we get:
$$
a_{n} = 0 quad mbox{ for } n geq 2,
$$
$$
a_{1} = 1,
$$
and
$$
a_{0} = lambda left(a_{0} + {a_{1} over 2}right).
$$
This gives a relationship between $a_{0}$ and $lambda$.
answered 7 hours ago
avsavs
3,769514
$endgroup$
add a comment |
$begingroup$
Note that since $lambdaint_0^1 phi(s),ds$ is a constant (with respect to $x$), then we can write$$phi(x)=x+a$$and by substitution we conclude that $$x+a=x+lambdaint _{0}^{1}x+adximplies\a=lambda({1over 2}+a)implies\a={lambdaover 2-2lambda}$$ and we obtain$$phi(x)=x+{lambdaover 2-2lambda}quad,quad lambdane 1$$The case $lambda=1$ leads to no solution.
answered 7 hours ago
Mostafa AyazMostafa Ayaz
17.6k31039
$endgroup$
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Relabelling the dummy variable $xmapsto s$ on the LHS of your final equation, $$int_0^1phi(s),ds-lambdaint_0^1phi(s),ds=frac12\implies int_0^1phi(s),ds=frac1{2(1-lambda)}$$
Thus $$phi(x)=x+fraclambda{2(1-lambda)}$$
answered 7 hours ago
John DoeJohn Doe
11.3k11239
$endgroup$
add a comment |
$begingroup$
Relabelling the dummy variable $xmapsto s$ on the LHS of your final equation, $$int_0^1phi(s),ds-lambdaint_0^1phi(s),ds=frac12\implies int_0^1phi(s),ds=frac1{2(1-lambda)}$$
Thus $$phi(x)=x+fraclambda{2(1-lambda)}$$
answered 7 hours ago
John DoeJohn Doe
11.3k11239
$endgroup$
add a comment |
$begingroup$
Relabelling the dummy variable $xmapsto s$ on the LHS of your final equation, $$int_0^1phi(s),ds-lambdaint_0^1phi(s),ds=frac12\implies int_0^1phi(s),ds=frac1{2(1-lambda)}$$
Thus $$phi(x)=x+fraclambda{2(1-lambda)}$$
answered 7 hours ago
John DoeJohn Doe
11.3k11239
$endgroup$
Relabelling the dummy variable $xmapsto s$ on the LHS of your final equation, $$int_0^1phi(s),ds-lambdaint_0^1phi(s),ds=frac12\implies int_0^1phi(s),ds=frac1{2(1-lambda)}$$
Thus $$phi(x)=x+fraclambda{2(1-lambda)}$$
answered 7 hours ago
John DoeJohn Doe
11.3k11239
answered 7 hours ago
John DoeJohn Doe
11.3k11239
answered 7 hours ago
John DoeJohn Doe
11.3k11239
answered 7 hours ago
John DoeJohn Doe
11.3k11239
11.3k11239
add a comment |
add a comment |
$begingroup$
Note $int_{0}^{1}{phi(s)ds}$ is a constant say $a$. Your functional equation (FE) can be rewritten as: $$phi(x)=x+alambda$$
Putting into FE yields:
$$x+alambda=x+lambdaint_{0}^{1}{(s+alambda )ds }iff alambda=lambdabig(frac{1}{2}+lambda abig)$$
If $lambda=0$ then $phi(x)=x$
if $lambdane 1$ $a=frac{1}{2}+lambda aiff ( 1-lambda)a=frac{1}{2}iff a=frac{1}{2-2lambda}$ and then $phi(x)=x+frac{lambda}{2-2lambda}$
If $lambda=1$ there won’t besuch $phi$.
answered 7 hours ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
1,478211
$endgroup$
add a comment |
$begingroup$
Note $int_{0}^{1}{phi(s)ds}$ is a constant say $a$. Your functional equation (FE) can be rewritten as: $$phi(x)=x+alambda$$
Putting into FE yields:
$$x+alambda=x+lambdaint_{0}^{1}{(s+alambda )ds }iff alambda=lambdabig(frac{1}{2}+lambda abig)$$
If $lambda=0$ then $phi(x)=x$
if $lambdane 1$ $a=frac{1}{2}+lambda aiff ( 1-lambda)a=frac{1}{2}iff a=frac{1}{2-2lambda}$ and then $phi(x)=x+frac{lambda}{2-2lambda}$
If $lambda=1$ there won’t besuch $phi$.
answered 7 hours ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
1,478211
$endgroup$
add a comment |
$begingroup$
Note $int_{0}^{1}{phi(s)ds}$ is a constant say $a$. Your functional equation (FE) can be rewritten as: $$phi(x)=x+alambda$$
Putting into FE yields:
$$x+alambda=x+lambdaint_{0}^{1}{(s+alambda )ds }iff alambda=lambdabig(frac{1}{2}+lambda abig)$$
If $lambda=0$ then $phi(x)=x$
if $lambdane 1$ $a=frac{1}{2}+lambda aiff ( 1-lambda)a=frac{1}{2}iff a=frac{1}{2-2lambda}$ and then $phi(x)=x+frac{lambda}{2-2lambda}$
If $lambda=1$ there won’t besuch $phi$.
answered 7 hours ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
1,478211
$endgroup$
Note $int_{0}^{1}{phi(s)ds}$ is a constant say $a$. Your functional equation (FE) can be rewritten as: $$phi(x)=x+alambda$$
Putting into FE yields:
$$x+alambda=x+lambdaint_{0}^{1}{(s+alambda )ds }iff alambda=lambdabig(frac{1}{2}+lambda abig)$$
If $lambda=0$ then $phi(x)=x$
if $lambdane 1$ $a=frac{1}{2}+lambda aiff ( 1-lambda)a=frac{1}{2}iff a=frac{1}{2-2lambda}$ and then $phi(x)=x+frac{lambda}{2-2lambda}$
If $lambda=1$ there won’t besuch $phi$.
answered 7 hours ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
1,478211
answered 7 hours ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
1,478211
answered 7 hours ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
1,478211
answered 7 hours ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
1,478211
1,478211
add a comment |
add a comment |
$begingroup$
If you are after finding $phi(x)$, one approach that comes to mind is to assume it is smooth enough to have a normally convergent (so we can interchange series summation and integration) Taylor expansion on $[0, 1]$:
$$
phi(x) = sum_{n geq 0} a_{n} x^{n}.
$$
Substituting it into your equation, we get:
$$
sum_{n geq 0} a_{n} x^{n} = x + lambda sum_{n geq 0}{a_{n} over n+1}.
$$
Matching up the coefficients of the difference powers of $x$, we get:
$$
a_{n} = 0 quad mbox{ for } n geq 2,
$$
$$
a_{1} = 1,
$$
and
$$
a_{0} = lambda left(a_{0} + {a_{1} over 2}right).
$$
This gives a relationship between $a_{0}$ and $lambda$.
answered 7 hours ago
avsavs
3,769514
$endgroup$
add a comment |
$begingroup$
If you are after finding $phi(x)$, one approach that comes to mind is to assume it is smooth enough to have a normally convergent (so we can interchange series summation and integration) Taylor expansion on $[0, 1]$:
$$
phi(x) = sum_{n geq 0} a_{n} x^{n}.
$$
Substituting it into your equation, we get:
$$
sum_{n geq 0} a_{n} x^{n} = x + lambda sum_{n geq 0}{a_{n} over n+1}.
$$
Matching up the coefficients of the difference powers of $x$, we get:
$$
a_{n} = 0 quad mbox{ for } n geq 2,
$$
$$
a_{1} = 1,
$$
and
$$
a_{0} = lambda left(a_{0} + {a_{1} over 2}right).
$$
This gives a relationship between $a_{0}$ and $lambda$.
answered 7 hours ago
avsavs
3,769514
$endgroup$
add a comment |
$begingroup$
If you are after finding $phi(x)$, one approach that comes to mind is to assume it is smooth enough to have a normally convergent (so we can interchange series summation and integration) Taylor expansion on $[0, 1]$:
$$
phi(x) = sum_{n geq 0} a_{n} x^{n}.
$$
Substituting it into your equation, we get:
$$
sum_{n geq 0} a_{n} x^{n} = x + lambda sum_{n geq 0}{a_{n} over n+1}.
$$
Matching up the coefficients of the difference powers of $x$, we get:
$$
a_{n} = 0 quad mbox{ for } n geq 2,
$$
$$
a_{1} = 1,
$$
and
$$
a_{0} = lambda left(a_{0} + {a_{1} over 2}right).
$$
This gives a relationship between $a_{0}$ and $lambda$.
answered 7 hours ago
avsavs
3,769514
$endgroup$
If you are after finding $phi(x)$, one approach that comes to mind is to assume it is smooth enough to have a normally convergent (so we can interchange series summation and integration) Taylor expansion on $[0, 1]$:
$$
phi(x) = sum_{n geq 0} a_{n} x^{n}.
$$
Substituting it into your equation, we get:
$$
sum_{n geq 0} a_{n} x^{n} = x + lambda sum_{n geq 0}{a_{n} over n+1}.
$$
Matching up the coefficients of the difference powers of $x$, we get:
$$
a_{n} = 0 quad mbox{ for } n geq 2,
$$
$$
a_{1} = 1,
$$
and
$$
a_{0} = lambda left(a_{0} + {a_{1} over 2}right).
$$
This gives a relationship between $a_{0}$ and $lambda$.
answered 7 hours ago
avsavs
3,769514
answered 7 hours ago
avsavs
3,769514
answered 7 hours ago
avsavs
3,769514
answered 7 hours ago
avsavs
3,769514
3,769514
add a comment |
add a comment |
$begingroup$
Note that since $lambdaint_0^1 phi(s),ds$ is a constant (with respect to $x$), then we can write$$phi(x)=x+a$$and by substitution we conclude that $$x+a=x+lambdaint _{0}^{1}x+adximplies\a=lambda({1over 2}+a)implies\a={lambdaover 2-2lambda}$$ and we obtain$$phi(x)=x+{lambdaover 2-2lambda}quad,quad lambdane 1$$The case $lambda=1$ leads to no solution.
answered 7 hours ago
Mostafa AyazMostafa Ayaz
17.6k31039
$endgroup$
add a comment |
$begingroup$
Note that since $lambdaint_0^1 phi(s),ds$ is a constant (with respect to $x$), then we can write$$phi(x)=x+a$$and by substitution we conclude that $$x+a=x+lambdaint _{0}^{1}x+adximplies\a=lambda({1over 2}+a)implies\a={lambdaover 2-2lambda}$$ and we obtain$$phi(x)=x+{lambdaover 2-2lambda}quad,quad lambdane 1$$The case $lambda=1$ leads to no solution.
answered 7 hours ago
Mostafa AyazMostafa Ayaz
17.6k31039
$endgroup$
add a comment |
$begingroup$
Note that since $lambdaint_0^1 phi(s),ds$ is a constant (with respect to $x$), then we can write$$phi(x)=x+a$$and by substitution we conclude that $$x+a=x+lambdaint _{0}^{1}x+adximplies\a=lambda({1over 2}+a)implies\a={lambdaover 2-2lambda}$$ and we obtain$$phi(x)=x+{lambdaover 2-2lambda}quad,quad lambdane 1$$The case $lambda=1$ leads to no solution.
answered 7 hours ago
Mostafa AyazMostafa Ayaz
17.6k31039
$endgroup$
Note that since $lambdaint_0^1 phi(s),ds$ is a constant (with respect to $x$), then we can write$$phi(x)=x+a$$and by substitution we conclude that $$x+a=x+lambdaint _{0}^{1}x+adximplies\a=lambda({1over 2}+a)implies\a={lambdaover 2-2lambda}$$ and we obtain$$phi(x)=x+{lambdaover 2-2lambda}quad,quad lambdane 1$$The case $lambda=1$ leads to no solution.
answered 7 hours ago
Mostafa AyazMostafa Ayaz
17.6k31039
answered 7 hours ago
Mostafa AyazMostafa Ayaz
17.6k31039
answered 7 hours ago
Mostafa AyazMostafa Ayaz
17.6k31039
answered 7 hours ago
Mostafa AyazMostafa Ayaz
17.6k31039
17.6k31039
add a comment |
add a comment |
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1
$begingroup$
What is $lambda$? What do you mean by "solve the problem"? I don't see what "the problem" is supposed to mean. Of which object do you want to find the characteristic values and ranks? Have you checked your definition of $phi$?
$endgroup$
– James
7 hours ago
$begingroup$
My apologies, $lambda$ is an arbitrary constant. In essence I want to obtain an expression of $phi(x)$ which does not contain a function of $s$, which the initial integral equation has.
$endgroup$
– LightningStrike
7 hours ago