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The solution of Abel's equation of the first kind
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Solving a problem I was faced with an Abel's equation of the first kind as
$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$ where $a,b$ and $k$ are nonzero constant and $W=b+asin kx$. I also know that $y(0)=1-b$.
I searched the Web and read some papers, however there are some solutions for some particular cases which do not contain my case.
Any ideas about finding either an analytical solution or numerical?
differential-equations
asked Jul 16 at 20:53
Majid
1,8521923
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up vote
0
down vote
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Solving a problem I was faced with an Abel's equation of the first kind as
$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$ where $a,b$ and $k$ are nonzero constant and $W=b+asin kx$. I also know that $y(0)=1-b$.
I searched the Web and read some papers, however there are some solutions for some particular cases which do not contain my case.
Any ideas about finding either an analytical solution or numerical?
differential-equations
asked Jul 16 at 20:53
Majid
1,8521923
sorry I forgot to write =0 :)
– Majid
Jul 17 at 13:44
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Solving a problem I was faced with an Abel's equation of the first kind as
$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$ where $a,b$ and $k$ are nonzero constant and $W=b+asin kx$. I also know that $y(0)=1-b$.
I searched the Web and read some papers, however there are some solutions for some particular cases which do not contain my case.
Any ideas about finding either an analytical solution or numerical?
differential-equations
asked Jul 16 at 20:53
Majid
1,8521923
Solving a problem I was faced with an Abel's equation of the first kind as
$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$ where $a,b$ and $k$ are nonzero constant and $W=b+asin kx$. I also know that $y(0)=1-b$.
I searched the Web and read some papers, however there are some solutions for some particular cases which do not contain my case.
Any ideas about finding either an analytical solution or numerical?
differential-equations
asked Jul 16 at 20:53
Majid
1,8521923
edited Jul 17 at 13:44
asked Jul 16 at 20:53
Majid
1,8521923
asked Jul 16 at 20:53
Majid
1,8521923
asked Jul 16 at 20:53
Majid
1,8521923
1,8521923
sorry I forgot to write =0 :)
– Majid
Jul 17 at 13:44
add a comment |Â
sorry I forgot to write =0 :)
– Majid
Jul 17 at 13:44
sorry I forgot to write =0 :)
– Majid
Jul 17 at 13:44
sorry I forgot to write =0 :)
– Majid
Jul 17 at 13:44
add a comment |Â
1 Answer
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$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$
$$y'-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$$
Change of variable: $quad t=asin(kx)+b-1,$
$fracdydx=fracdydtfracdtdx=akcos(kx)fracdydt$
$akcos(kx)fracdydt-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$
$$fracdydt-y^3-t=0$$
Of course, this is still an Abel's ODE, but much simpler than the original one : There is no parameter in it.
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
answered Jul 17 at 7:13
JJacquelin
40.1k21649
JJacquelin this is the canonical form of Abel. Right? There are some papers that gave some solutions for some cases of Abel. I am wondering if there is not anything related to this one. As you mentioned it is much simpler now
– Majid
Jul 17 at 14:00
@Majid.I suggest this paper : arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf . Sorry, I have not enough available time just now. Good luck.
– JJacquelin
Jul 17 at 14:55
Thank you for your time! In this paper a solution is suggested for a restricted first kind of Abel. Moreover I do not know how to write this equation as the Eq. (3) suggested in this paper. Any suggestion?
– Majid
Jul 17 at 19:47
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
– JJacquelin
Jul 19 at 10:41
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$
$$y'-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$$
Change of variable: $quad t=asin(kx)+b-1,$
$fracdydx=fracdydtfracdtdx=akcos(kx)fracdydt$
$akcos(kx)fracdydt-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$
$$fracdydt-y^3-t=0$$
Of course, this is still an Abel's ODE, but much simpler than the original one : There is no parameter in it.
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
answered Jul 17 at 7:13
JJacquelin
40.1k21649
JJacquelin this is the canonical form of Abel. Right? There are some papers that gave some solutions for some cases of Abel. I am wondering if there is not anything related to this one. As you mentioned it is much simpler now
– Majid
Jul 17 at 14:00
@Majid.I suggest this paper : arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf . Sorry, I have not enough available time just now. Good luck.
– JJacquelin
Jul 17 at 14:55
Thank you for your time! In this paper a solution is suggested for a restricted first kind of Abel. Moreover I do not know how to write this equation as the Eq. (3) suggested in this paper. Any suggestion?
– Majid
Jul 17 at 19:47
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
– JJacquelin
Jul 19 at 10:41
add a comment |Â
up vote
2
down vote
$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$
$$y'-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$$
Change of variable: $quad t=asin(kx)+b-1,$
$fracdydx=fracdydtfracdtdx=akcos(kx)fracdydt$
$akcos(kx)fracdydt-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$
$$fracdydt-y^3-t=0$$
Of course, this is still an Abel's ODE, but much simpler than the original one : There is no parameter in it.
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
answered Jul 17 at 7:13
JJacquelin
40.1k21649
JJacquelin this is the canonical form of Abel. Right? There are some papers that gave some solutions for some cases of Abel. I am wondering if there is not anything related to this one. As you mentioned it is much simpler now
– Majid
Jul 17 at 14:00
@Majid.I suggest this paper : arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf . Sorry, I have not enough available time just now. Good luck.
– JJacquelin
Jul 17 at 14:55
Thank you for your time! In this paper a solution is suggested for a restricted first kind of Abel. Moreover I do not know how to write this equation as the Eq. (3) suggested in this paper. Any suggestion?
– Majid
Jul 17 at 19:47
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
– JJacquelin
Jul 19 at 10:41
add a comment |Â
up vote
2
down vote
up vote
2
down vote
$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$
$$y'-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$$
Change of variable: $quad t=asin(kx)+b-1,$
$fracdydx=fracdydtfracdtdx=akcos(kx)fracdydt$
$akcos(kx)fracdydt-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$
$$fracdydt-y^3-t=0$$
Of course, this is still an Abel's ODE, but much simpler than the original one : There is no parameter in it.
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
answered Jul 17 at 7:13
JJacquelin
40.1k21649
$$y'-(akcos kx)y^3+(1-W)akcos kx=0,$$
$$y'-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$$
Change of variable: $quad t=asin(kx)+b-1,$
$fracdydx=fracdydtfracdtdx=akcos(kx)fracdydt$
$akcos(kx)fracdydt-akcos(kx)y^3+(1-b-asin(kx))akcos(kx)=0$
$$fracdydt-y^3-t=0$$
Of course, this is still an Abel's ODE, but much simpler than the original one : There is no parameter in it.
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
answered Jul 17 at 7:13
JJacquelin
40.1k21649
edited Jul 17 at 13:49
answered Jul 17 at 7:13
JJacquelin
40.1k21649
answered Jul 17 at 7:13
JJacquelin
40.1k21649
answered Jul 17 at 7:13
JJacquelin
40.1k21649
40.1k21649
JJacquelin this is the canonical form of Abel. Right? There are some papers that gave some solutions for some cases of Abel. I am wondering if there is not anything related to this one. As you mentioned it is much simpler now
– Majid
Jul 17 at 14:00
@Majid.I suggest this paper : arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf . Sorry, I have not enough available time just now. Good luck.
– JJacquelin
Jul 17 at 14:55
Thank you for your time! In this paper a solution is suggested for a restricted first kind of Abel. Moreover I do not know how to write this equation as the Eq. (3) suggested in this paper. Any suggestion?
– Majid
Jul 17 at 19:47
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
– JJacquelin
Jul 19 at 10:41
add a comment |Â
JJacquelin this is the canonical form of Abel. Right? There are some papers that gave some solutions for some cases of Abel. I am wondering if there is not anything related to this one. As you mentioned it is much simpler now
– Majid
Jul 17 at 14:00
@Majid.I suggest this paper : arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf . Sorry, I have not enough available time just now. Good luck.
– JJacquelin
Jul 17 at 14:55
Thank you for your time! In this paper a solution is suggested for a restricted first kind of Abel. Moreover I do not know how to write this equation as the Eq. (3) suggested in this paper. Any suggestion?
– Majid
Jul 17 at 19:47
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
– JJacquelin
Jul 19 at 10:41
JJacquelin this is the canonical form of Abel. Right? There are some papers that gave some solutions for some cases of Abel. I am wondering if there is not anything related to this one. As you mentioned it is much simpler now
– Majid
Jul 17 at 14:00
JJacquelin this is the canonical form of Abel. Right? There are some papers that gave some solutions for some cases of Abel. I am wondering if there is not anything related to this one. As you mentioned it is much simpler now
– Majid
Jul 17 at 14:00
@Majid.I suggest this paper : arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf . Sorry, I have not enough available time just now. Good luck.
– JJacquelin
Jul 17 at 14:55
@Majid.I suggest this paper : arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf . Sorry, I have not enough available time just now. Good luck.
– JJacquelin
Jul 17 at 14:55
Thank you for your time! In this paper a solution is suggested for a restricted first kind of Abel. Moreover I do not know how to write this equation as the Eq. (3) suggested in this paper. Any suggestion?
– Majid
Jul 17 at 19:47
Thank you for your time! In this paper a solution is suggested for a restricted first kind of Abel. Moreover I do not know how to write this equation as the Eq. (3) suggested in this paper. Any suggestion?
– Majid
Jul 17 at 19:47
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
– JJacquelin
Jul 19 at 10:41
Apparently, this is not a solvable case in terms of standard special functions. So, it is suggested to continue thanks to numerical method.
– JJacquelin
Jul 19 at 10:41
add a comment |Â
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sorry I forgot to write =0 :)
– Majid
Jul 17 at 13:44