difference between linear, semilinear and quasiliner PDE's












3












$begingroup$


I know a PDE is linear when the dependent variable $u$ and its derivatives appear only to the first power. So, $u_t + u_x +5u = 1$ would be linear. However, I do not quite understand the other two.



My professor described "semilinear" PDE's as PDE's whose highest order terms are linear, and "quasilinear" PDE's as PDE's whose highest order terms appear only as individual terms multiplied by lower order terms. No examples were provided, only equivalent statements involving sums and multiindeces which I do not think I could decipher by tomorrow.



Can someone provide some examples of "semilinear" and "quasilinear" PDE's?






pde







share|cite|improve this question









$endgroup$












  • $begingroup$
    You can find a formal definition in 'Partial Differential Equations': Second Edition written by Evans. It is on the second page of chapter 1.
    $endgroup$
    – esmo
    Jul 29 '18 at 9:11










  • $begingroup$
    Another good reference book is from the book of Zachmanoglou"introduction to PDE with applications"
    $endgroup$
    – dmtri
    Jul 29 '18 at 9:32


















3












$begingroup$


I know a PDE is linear when the dependent variable $u$ and its derivatives appear only to the first power. So, $u_t + u_x +5u = 1$ would be linear. However, I do not quite understand the other two.



My professor described "semilinear" PDE's as PDE's whose highest order terms are linear, and "quasilinear" PDE's as PDE's whose highest order terms appear only as individual terms multiplied by lower order terms. No examples were provided, only equivalent statements involving sums and multiindeces which I do not think I could decipher by tomorrow.



Can someone provide some examples of "semilinear" and "quasilinear" PDE's?






pde







share|cite|improve this question









$endgroup$












  • $begingroup$
    You can find a formal definition in 'Partial Differential Equations': Second Edition written by Evans. It is on the second page of chapter 1.
    $endgroup$
    – esmo
    Jul 29 '18 at 9:11










  • $begingroup$
    Another good reference book is from the book of Zachmanoglou"introduction to PDE with applications"
    $endgroup$
    – dmtri
    Jul 29 '18 at 9:32
















3












3








3


0



$begingroup$


I know a PDE is linear when the dependent variable $u$ and its derivatives appear only to the first power. So, $u_t + u_x +5u = 1$ would be linear. However, I do not quite understand the other two.



My professor described "semilinear" PDE's as PDE's whose highest order terms are linear, and "quasilinear" PDE's as PDE's whose highest order terms appear only as individual terms multiplied by lower order terms. No examples were provided, only equivalent statements involving sums and multiindeces which I do not think I could decipher by tomorrow.



Can someone provide some examples of "semilinear" and "quasilinear" PDE's?






pde







share|cite|improve this question









$endgroup$




I know a PDE is linear when the dependent variable $u$ and its derivatives appear only to the first power. So, $u_t + u_x +5u = 1$ would be linear. However, I do not quite understand the other two.



My professor described "semilinear" PDE's as PDE's whose highest order terms are linear, and "quasilinear" PDE's as PDE's whose highest order terms appear only as individual terms multiplied by lower order terms. No examples were provided, only equivalent statements involving sums and multiindeces which I do not think I could decipher by tomorrow.



Can someone provide some examples of "semilinear" and "quasilinear" PDE's?






pde




pde






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 25 '17 at 3:55









Ninosław CiszewskiNinosław Ciszewski

531412




531412












  • $begingroup$
    You can find a formal definition in 'Partial Differential Equations': Second Edition written by Evans. It is on the second page of chapter 1.
    $endgroup$
    – esmo
    Jul 29 '18 at 9:11










  • $begingroup$
    Another good reference book is from the book of Zachmanoglou"introduction to PDE with applications"
    $endgroup$
    – dmtri
    Jul 29 '18 at 9:32




















  • $begingroup$
    You can find a formal definition in 'Partial Differential Equations': Second Edition written by Evans. It is on the second page of chapter 1.
    $endgroup$
    – esmo
    Jul 29 '18 at 9:11










  • $begingroup$
    Another good reference book is from the book of Zachmanoglou"introduction to PDE with applications"
    $endgroup$
    – dmtri
    Jul 29 '18 at 9:32


















$begingroup$
You can find a formal definition in 'Partial Differential Equations': Second Edition written by Evans. It is on the second page of chapter 1.
$endgroup$
– esmo
Jul 29 '18 at 9:11




$begingroup$
You can find a formal definition in 'Partial Differential Equations': Second Edition written by Evans. It is on the second page of chapter 1.
$endgroup$
– esmo
Jul 29 '18 at 9:11












$begingroup$
Another good reference book is from the book of Zachmanoglou"introduction to PDE with applications"
$endgroup$
– dmtri
Jul 29 '18 at 9:32






$begingroup$
Another good reference book is from the book of Zachmanoglou"introduction to PDE with applications"
$endgroup$
– dmtri
Jul 29 '18 at 9:32












2 Answers
2






active

oldest

votes


















6












$begingroup$

I think this will help you to understand the PDE $:$



Linear PDE: $a(x,y)u_x+b(x,y)u_y+c(x,y)u=f(x,y)$



Semi-linear PDE: $a(x,y)u_x+b(x,y)u_y=f(x,y,u)$



Quasi-linear PDE: $a(x,y,u)u_x+b(x,y,u)u_y=f(x,y,u)$






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    could you please give a similar explanation for the case of second order pde ?
    $endgroup$
    – Nizar
    Sep 30 '18 at 12:38










  • $begingroup$
    @MatheMagic thanks for your answer.I have one query.You consider first order PDEs in your answer,can I know how coefficients will change if it is of second order means suppose in case of quasi-linear $u_{xx}$ is present then coefficient will remain a(x,y,u) or will get modify?
    $endgroup$
    – Believer
    2 days ago



















-1












$begingroup$

I hope these examples will help you.



Semilinear/Almost Linear PDE:



1) $a(x,y)u_x+b(x,y)u_y+c(x,y,u)=0$



2) $U_{tt}-U_{xx}+U^3=0$



Qausi Linear PDE:



1) $a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0$



2) $U_x+UV_y=0$



3) $U_{tt}-UU_{xx}+U^3=0$



4) $U_{tt}-UU_{xx}+U=0$



5) Navier Stokes equation is also Qausi Linear Equation






share|cite|improve this answer











$endgroup$













    Your Answer





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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    6












    $begingroup$

    I think this will help you to understand the PDE $:$



    Linear PDE: $a(x,y)u_x+b(x,y)u_y+c(x,y)u=f(x,y)$



    Semi-linear PDE: $a(x,y)u_x+b(x,y)u_y=f(x,y,u)$



    Quasi-linear PDE: $a(x,y,u)u_x+b(x,y,u)u_y=f(x,y,u)$






    share|cite|improve this answer









    $endgroup$









    • 1




      $begingroup$
      could you please give a similar explanation for the case of second order pde ?
      $endgroup$
      – Nizar
      Sep 30 '18 at 12:38










    • $begingroup$
      @MatheMagic thanks for your answer.I have one query.You consider first order PDEs in your answer,can I know how coefficients will change if it is of second order means suppose in case of quasi-linear $u_{xx}$ is present then coefficient will remain a(x,y,u) or will get modify?
      $endgroup$
      – Believer
      2 days ago
















    6












    $begingroup$

    I think this will help you to understand the PDE $:$



    Linear PDE: $a(x,y)u_x+b(x,y)u_y+c(x,y)u=f(x,y)$



    Semi-linear PDE: $a(x,y)u_x+b(x,y)u_y=f(x,y,u)$



    Quasi-linear PDE: $a(x,y,u)u_x+b(x,y,u)u_y=f(x,y,u)$






    share|cite|improve this answer









    $endgroup$









    • 1




      $begingroup$
      could you please give a similar explanation for the case of second order pde ?
      $endgroup$
      – Nizar
      Sep 30 '18 at 12:38










    • $begingroup$
      @MatheMagic thanks for your answer.I have one query.You consider first order PDEs in your answer,can I know how coefficients will change if it is of second order means suppose in case of quasi-linear $u_{xx}$ is present then coefficient will remain a(x,y,u) or will get modify?
      $endgroup$
      – Believer
      2 days ago














    6












    6








    6





    $begingroup$

    I think this will help you to understand the PDE $:$



    Linear PDE: $a(x,y)u_x+b(x,y)u_y+c(x,y)u=f(x,y)$



    Semi-linear PDE: $a(x,y)u_x+b(x,y)u_y=f(x,y,u)$



    Quasi-linear PDE: $a(x,y,u)u_x+b(x,y,u)u_y=f(x,y,u)$






    share|cite|improve this answer









    $endgroup$



    I think this will help you to understand the PDE $:$



    Linear PDE: $a(x,y)u_x+b(x,y)u_y+c(x,y)u=f(x,y)$



    Semi-linear PDE: $a(x,y)u_x+b(x,y)u_y=f(x,y,u)$



    Quasi-linear PDE: $a(x,y,u)u_x+b(x,y,u)u_y=f(x,y,u)$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Jan 25 '17 at 4:40









    MatheMagicMatheMagic

    1,3621617




    1,3621617








    • 1




      $begingroup$
      could you please give a similar explanation for the case of second order pde ?
      $endgroup$
      – Nizar
      Sep 30 '18 at 12:38










    • $begingroup$
      @MatheMagic thanks for your answer.I have one query.You consider first order PDEs in your answer,can I know how coefficients will change if it is of second order means suppose in case of quasi-linear $u_{xx}$ is present then coefficient will remain a(x,y,u) or will get modify?
      $endgroup$
      – Believer
      2 days ago














    • 1




      $begingroup$
      could you please give a similar explanation for the case of second order pde ?
      $endgroup$
      – Nizar
      Sep 30 '18 at 12:38










    • $begingroup$
      @MatheMagic thanks for your answer.I have one query.You consider first order PDEs in your answer,can I know how coefficients will change if it is of second order means suppose in case of quasi-linear $u_{xx}$ is present then coefficient will remain a(x,y,u) or will get modify?
      $endgroup$
      – Believer
      2 days ago








    1




    1




    $begingroup$
    could you please give a similar explanation for the case of second order pde ?
    $endgroup$
    – Nizar
    Sep 30 '18 at 12:38




    $begingroup$
    could you please give a similar explanation for the case of second order pde ?
    $endgroup$
    – Nizar
    Sep 30 '18 at 12:38












    $begingroup$
    @MatheMagic thanks for your answer.I have one query.You consider first order PDEs in your answer,can I know how coefficients will change if it is of second order means suppose in case of quasi-linear $u_{xx}$ is present then coefficient will remain a(x,y,u) or will get modify?
    $endgroup$
    – Believer
    2 days ago




    $begingroup$
    @MatheMagic thanks for your answer.I have one query.You consider first order PDEs in your answer,can I know how coefficients will change if it is of second order means suppose in case of quasi-linear $u_{xx}$ is present then coefficient will remain a(x,y,u) or will get modify?
    $endgroup$
    – Believer
    2 days ago











    -1












    $begingroup$

    I hope these examples will help you.



    Semilinear/Almost Linear PDE:



    1) $a(x,y)u_x+b(x,y)u_y+c(x,y,u)=0$



    2) $U_{tt}-U_{xx}+U^3=0$



    Qausi Linear PDE:



    1) $a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0$



    2) $U_x+UV_y=0$



    3) $U_{tt}-UU_{xx}+U^3=0$



    4) $U_{tt}-UU_{xx}+U=0$



    5) Navier Stokes equation is also Qausi Linear Equation






    share|cite|improve this answer











    $endgroup$


















      -1












      $begingroup$

      I hope these examples will help you.



      Semilinear/Almost Linear PDE:



      1) $a(x,y)u_x+b(x,y)u_y+c(x,y,u)=0$



      2) $U_{tt}-U_{xx}+U^3=0$



      Qausi Linear PDE:



      1) $a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0$



      2) $U_x+UV_y=0$



      3) $U_{tt}-UU_{xx}+U^3=0$



      4) $U_{tt}-UU_{xx}+U=0$



      5) Navier Stokes equation is also Qausi Linear Equation






      share|cite|improve this answer











      $endgroup$
















        -1












        -1








        -1





        $begingroup$

        I hope these examples will help you.



        Semilinear/Almost Linear PDE:



        1) $a(x,y)u_x+b(x,y)u_y+c(x,y,u)=0$



        2) $U_{tt}-U_{xx}+U^3=0$



        Qausi Linear PDE:



        1) $a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0$



        2) $U_x+UV_y=0$



        3) $U_{tt}-UU_{xx}+U^3=0$



        4) $U_{tt}-UU_{xx}+U=0$



        5) Navier Stokes equation is also Qausi Linear Equation






        share|cite|improve this answer











        $endgroup$



        I hope these examples will help you.



        Semilinear/Almost Linear PDE:



        1) $a(x,y)u_x+b(x,y)u_y+c(x,y,u)=0$



        2) $U_{tt}-U_{xx}+U^3=0$



        Qausi Linear PDE:



        1) $a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0$



        2) $U_x+UV_y=0$



        3) $U_{tt}-UU_{xx}+U^3=0$



        4) $U_{tt}-UU_{xx}+U=0$



        5) Navier Stokes equation is also Qausi Linear Equation







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 8 '18 at 10:36









        Brahadeesh

        6,19742361




        6,19742361










        answered Dec 8 '18 at 9:00









        Hammad KhalidHammad Khalid

        12




        12






























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