confluent differential equation
$begingroup$
I have a similar problem and I would appreciate any help in this matter. The DE that I am dealing with is:
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
where $a, b, c$ are constant and $f_z$ is the first derivative of $f(z)$ and $f_{zz}$ is the second derivative.
I believe that this eq. must be related to the confleunt differential equation, but for some reasons I cannot find this form in the books. Can anyone give a reference books where I can find this equation and its solution? Many thanks.
ordinary-differential-equations
edited Jan 10 at 20:26
greelious
472112
asked Jan 10 at 18:58
user630770user630770
6
$endgroup$
add a comment |
$begingroup$
I have a similar problem and I would appreciate any help in this matter. The DE that I am dealing with is:
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
where $a, b, c$ are constant and $f_z$ is the first derivative of $f(z)$ and $f_{zz}$ is the second derivative.
I believe that this eq. must be related to the confleunt differential equation, but for some reasons I cannot find this form in the books. Can anyone give a reference books where I can find this equation and its solution? Many thanks.
ordinary-differential-equations
edited Jan 10 at 20:26
greelious
472112
asked Jan 10 at 18:58
user630770user630770
6
$endgroup$
add a comment |
$begingroup$
I have a similar problem and I would appreciate any help in this matter. The DE that I am dealing with is:
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
where $a, b, c$ are constant and $f_z$ is the first derivative of $f(z)$ and $f_{zz}$ is the second derivative.
I believe that this eq. must be related to the confleunt differential equation, but for some reasons I cannot find this form in the books. Can anyone give a reference books where I can find this equation and its solution? Many thanks.
ordinary-differential-equations
edited Jan 10 at 20:26
greelious
472112
asked Jan 10 at 18:58
user630770user630770
6
$endgroup$
I have a similar problem and I would appreciate any help in this matter. The DE that I am dealing with is:
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
where $a, b, c$ are constant and $f_z$ is the first derivative of $f(z)$ and $f_{zz}$ is the second derivative.
I believe that this eq. must be related to the confleunt differential equation, but for some reasons I cannot find this form in the books. Can anyone give a reference books where I can find this equation and its solution? Many thanks.
ordinary-differential-equations
ordinary-differential-equations
edited Jan 10 at 20:26
greelious
472112
asked Jan 10 at 18:58
user630770user630770
6
edited Jan 10 at 20:26
greelious
472112
asked Jan 10 at 18:58
user630770user630770
6
edited Jan 10 at 20:26
greelious
472112
edited Jan 10 at 20:26
greelious
472112
edited Jan 10 at 20:26
greelious
472112
472112
asked Jan 10 at 18:58
user630770user630770
6
asked Jan 10 at 18:58
user630770user630770
6
asked Jan 10 at 18:58
user630770user630770
6
6
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :
Confluent : $zf''+(c-x)f'-af=0$
hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$
The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.
I am afraid that you will have to use numerical methods of solving.
answered Jan 11 at 11:50
JJacquelinJJacquelin
44.3k21854
$endgroup$
add a comment |
$begingroup$
In fact this relates to Heun's Confluent Equation.
answered Jan 11 at 12:36
doraemonpauldoraemonpaul
12.7k31660
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :
Confluent : $zf''+(c-x)f'-af=0$
hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$
The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.
I am afraid that you will have to use numerical methods of solving.
answered Jan 11 at 11:50
JJacquelinJJacquelin
44.3k21854
$endgroup$
add a comment |
$begingroup$
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :
Confluent : $zf''+(c-x)f'-af=0$
hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$
The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.
I am afraid that you will have to use numerical methods of solving.
answered Jan 11 at 11:50
JJacquelinJJacquelin
44.3k21854
$endgroup$
add a comment |
$begingroup$
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :
Confluent : $zf''+(c-x)f'-af=0$
hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$
The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.
I am afraid that you will have to use numerical methods of solving.
answered Jan 11 at 11:50
JJacquelinJJacquelin
44.3k21854
$endgroup$
$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :
Confluent : $zf''+(c-x)f'-af=0$
hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$
The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.
I am afraid that you will have to use numerical methods of solving.
answered Jan 11 at 11:50
JJacquelinJJacquelin
44.3k21854
answered Jan 11 at 11:50
JJacquelinJJacquelin
44.3k21854
answered Jan 11 at 11:50
JJacquelinJJacquelin
44.3k21854
answered Jan 11 at 11:50
JJacquelinJJacquelin
44.3k21854
44.3k21854
add a comment |
add a comment |
$begingroup$
In fact this relates to Heun's Confluent Equation.
answered Jan 11 at 12:36
doraemonpauldoraemonpaul
12.7k31660
$endgroup$
add a comment |
$begingroup$
In fact this relates to Heun's Confluent Equation.
answered Jan 11 at 12:36
doraemonpauldoraemonpaul
12.7k31660
$endgroup$
add a comment |
$begingroup$
In fact this relates to Heun's Confluent Equation.
answered Jan 11 at 12:36
doraemonpauldoraemonpaul
12.7k31660
$endgroup$
In fact this relates to Heun's Confluent Equation.
answered Jan 11 at 12:36
doraemonpauldoraemonpaul
12.7k31660
answered Jan 11 at 12:36
doraemonpauldoraemonpaul
12.7k31660
answered Jan 11 at 12:36
doraemonpauldoraemonpaul
12.7k31660
answered Jan 11 at 12:36
doraemonpauldoraemonpaul
12.7k31660
12.7k31660
add a comment |
add a comment |
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