Solve the following system of partial differential equations { ∂ ∂ a S ( a , b , c , d ) = f 1 ( a ) ∂ ∂ b S ( a , b , c , d ) = f 2 ( b ) ∂ ∂ c S ( a , b , c , d ) = f 3 ( c ) ∂ ∂ d S ( a , b , c , d ) = f 4 ( d ) where f i ( )s are some nonlinear functions.Does the above system have a unique answer? And if has can any one introduce a reference, explaining the techniques for analytic solutions?

Question

Solve the following system of partial differential equations                              {                                                                      ∂                                      ∂                    a                                                  S                (                a                ,                b                ,                c                ,                d                )                =                                  f                  1                                (                a                )                                                                                      ∂                                      ∂                    b                                                  S                (                a                ,                b                ,                c                ,                d                )                =                                  f                  2                                (                b                )                                                                                      ∂                                      ∂                    c                                                  S                (                a                ,                b                ,                c                ,                d                )                =                                  f                  3                                (                c                )                                                                                      ∂                                      ∂                    d                                                  S                (                a                ,                b                ,                c                ,                d                )                =                                  f                  4                                (                d                )                                                                  where       f    i    (  )s are some nonlinear functions.Does the above system have a unique answer? And if has can any one introduce a reference, explaining the techniques for analytic solutions?

Zoe Andersen · Accepted Answer

Your problem can be reformulated as follows (upon the change of notation   a  =  x  ,  b  =  y  ,  c  =  z  ,  d  =  t).Assigning the differential form  ω  =      f    1    (  x  )  d  x  +      f    2    (  y  )  d  y  +      f    3    (  z  )  d  z  +      f    4    (  t  )  d  t  ,which is closed, hence exact on             R        4  . You want to find a potential function, that is, a function   F  =  F  (  x  ,  y  ,  z  ,  t  ) such that   d  F  =  ω. One of them is given by the following integral, the others differ by an additive constant:  F  (  x  ,  y  ,  z  ,  t  )  =      ∫    γ    ω  ,    γ  :  [  0  ,  1  ]  →            R        4    ,     γ  (  0  )  =  0  ,     γ  (  1  )  =  (  x  ,  y  ,  z  ,  t  )You can choose any curve   γ, only its endpoints matter. Taking for instance the line segment  γ  (  s  )  =  (  s  x  ,  s  y  ,  s  z  ,  s  t  )  ,you obtain  F  (  x  ,  y  ,  z  ,  t  )  =      ∫    0    1        f    1    (  s  x  )  x    d  s  +      ∫    0    1        f    2    (  s  y  )  y    d  s  +      ∫    0    1        f    3    (  s  z  )  z    d  s  +      ∫    0    1        f    4    (  s  t  )  t    d  s  .

Solve the following system of partial differential equations {d/da S(a,b,c,d)=f_1(a), d/db S(a,b,c,d)=f_2(b), d/dc S(a,b,c,d)=f_3(c), d/dd S(a,b,c,d)=f_4(d), where f_i()s are some nonlinear functions. Does the above system have a unique answer?

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