Solve the following initial value problem: u t + u x = v , v t + v x = − u , u ( 0 , x ) = u 0 ( x ) , v ( 0 , x ) = v 0 ( x ) .

Question

Solve the following initial value problem:      u    t    +      u    x    =  v  ,        v    t    +      v    x    =  −  u  ,    u  (  0  ,  x  )  =      u    0    (  x  )  ,    v  (  0  ,  x  )  =      v    0    (  x  )  .

trantegisis · Accepted Answer

u          t      t        +      u          x      t        =      v    t        u          x      t        +      u          x      x        =      v    x    ∴      v    t    +      v    x    =      u          t      t        +  2      u          x      t        +      u          x      x        =  −  u      u          t      t        +  2      u          x      t        +      u          x      x        +  u  =  0Let       {                            p          =          x          +          t                                      q          =          x          −          t                        Then                     ∂        u                    ∂        x              =                    ∂        u                    ∂        p                                ∂        p                    ∂        x              +                    ∂        u                    ∂        q                                ∂        q                    ∂        x              =                    ∂        u                    ∂        p              +                    ∂        u                    ∂        q                                          ∂          2                u                    ∂                  x          2                      =            ∂              ∂        x                  (                            ∂          u                          ∂          p                      +                            ∂          u                          ∂          q                      )    =            ∂              ∂        p                  (                            ∂          u                          ∂          p                      +                            ∂          u                          ∂          q                      )                      ∂        p                    ∂        x              +            ∂              ∂        q                  (                            ∂          u                          ∂          p                      +                            ∂          u                          ∂          q                      )                      ∂        q                    ∂        x              =                              ∂          2                u                    ∂                  p          2                      +                              ∂          2                u                    ∂        p        q              +                              ∂          2                u                    ∂        p        q              +                              ∂          2                u                    ∂                  q          2                      =                              ∂          2                u                    ∂                  p          2                      +  2                              ∂          2                u                    ∂        p        q              +                              ∂          2                u                    ∂                  q          2                                        ∂        u                    ∂        t              =                    ∂        u                    ∂        p                                ∂        p                    ∂        t              +                    ∂        u                    ∂        q                                ∂        q                    ∂        t              =                    ∂        u                    ∂        p              −                    ∂        u                    ∂        q                                          ∂          2                u                    ∂        x        ∂        t              =            ∂              ∂        x                  (                            ∂          u                          ∂          p                      −                            ∂          u                          ∂          q                      )    =            ∂              ∂        p                  (                            ∂          u                          ∂          p                      −                            ∂          u                          ∂          q                      )                      ∂        p                    ∂        x              +            ∂              ∂        q                  (                            ∂          u                          ∂          p                      −                            ∂          u                          ∂          q                      )                      ∂        q                    ∂        x              =                              ∂          2                u                    ∂                  p          2                      −                              ∂          2                u                    ∂        p        q              +                              ∂          2                u                    ∂        p        q              −                              ∂          2                u                    ∂                  q          2                      =                              ∂          2                u                    ∂                  p          2                      −                              ∂          2                u                    ∂                  q          2                      ∴                              ∂          2                u                    ∂                  p          2                      −  2                              ∂          2                u                    ∂        p        q              +                              ∂          2                u                    ∂                  q          2                      +  2      (                                        ∂            2                    u                          ∂                      p            2                                −                                        ∂            2                    u                          ∂                      q            2                                )    +                              ∂          2                u                    ∂                  p          2                      +  2                              ∂          2                u                    ∂        p        q              +                              ∂          2                u                    ∂                  q          2                      +  u  =  0  4                              ∂          2                u                    ∂                  p          2                      +  u  =  0  u  (  p  ,  q  )  =  f  (  q  )  sin  ⁡            p      2        +  g  (  q  )  cos  ⁡            p      2        u  (  t  ,  x  )  =  f  (  x  −  t  )  sin  ⁡                    x        +        t            2        +  g  (  x  −  t  )  cos  ⁡                    x        +        t            2            u    t    (  t  ,  x  )  =  −      f    t    (  x  −  t  )  sin  ⁡                    x        +        t            2        +                    f        (        x        −        t        )            2        cos  ⁡                    x        +        t            2        −      g    t    (  x  −  t  )  cos  ⁡                    x        +        t            2        −                    g        (        x        −        t        )            2        sin  ⁡                    x        +        t            2            u    x    (  t  ,  x  )  =      f    x    (  x  −  t  )  sin  ⁡                    x        +        t            2        +                    f        (        x        −        t        )            2        cos  ⁡                    x        +        t            2        +      g    x    (  x  −  t  )  cos  ⁡                    x        +        t            2        −                    g        (        x        −        t        )            2        sin  ⁡                    x        +        t            2        ∴  v  (  t  ,  x  )  =      f          x      −      t        (  x  −  t  )  sin  ⁡                    x        +        t            2        +      g          x      −      t        (  x  −  t  )  cos  ⁡                    x        +        t            2        +  f  (  x  −  t  )  cos  ⁡                    x        +        t            2        −  g  (  x  −  t  )  sin  ⁡                    x        +        t            2      Hence      {                            u          (          t          ,          x          )          =          f          (          x          −          t          )          sin          ⁡                                                    x                +                t                            2                                +          g          (          x          −          t          )          cos          ⁡                                                    x                +                t                            2                                                            v          (          t          ,          x          )          =                      f                          x              −              t                                (          x          −          t          )          sin          ⁡                                                    x                +                t                            2                                +                      g                          x              −              t                                (          x          −          t          )          cos          ⁡                                                    x                +                t                            2                                +          f          (          x          −          t          )          cos          ⁡                                                    x                +                t                            2                                −          g          (          x          −          t          )          sin          ⁡                                                    x                +                t                            2

Solve the following initial value problem: u t </msub> + u x

Answered question

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