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Introduction Classifications Canonical forms Separation of variables Introduction to Partial Differential Equations Introductory Course on Multiphysics Modelling T OMASZ G. Z IELI NSKI (after: S.J. FARLOWs Partial Differential

  1. Introduction Classifications Canonical forms Separation of variables Well-posed and ill-posed problems Definition (A well-posed problem) An initial-boundary-value problem is well-posed if: 1 it has a unique solution ,

  2. Introduction Classifications Canonical forms Separation of variables Well-posed and ill-posed problems Definition (A well-posed problem) An initial-boundary-value problem is well-posed if: 1 it has a unique solution , 2 the solution vary continuously with the given inhomogeneous data, that is, small changes in the data should cause only small changes in the solution .

  3. Introduction Classifications Canonical forms Separation of variables Well-posed and ill-posed problems Definition (A well-posed problem) An initial-boundary-value problem is well-posed if: 1 it has a unique solution , 2 the solution vary continuously with the given inhomogeneous data, that is, small changes in the data should cause only small changes in the solution . Importance of well-posedness: In practice, the initial and boundary data are measured and so small errors occur.

  4. Introduction Classifications Canonical forms Separation of variables Well-posed and ill-posed problems Definition (A well-posed problem) An initial-boundary-value problem is well-posed if: 1 it has a unique solution , 2 the solution vary continuously with the given inhomogeneous data, that is, small changes in the data should cause only small changes in the solution . Importance of well-posedness: In practice, the initial and boundary data are measured and so small errors occur. Very often the problem must be solved numerically which involves truncation and round-off errors.

  5. Introduction Classifications Canonical forms Separation of variables Well-posed and ill-posed problems Definition (A well-posed problem) An initial-boundary-value problem is well-posed if: 1 it has a unique solution , 2 the solution vary continuously with the given inhomogeneous data, that is, small changes in the data should cause only small changes in the solution . Importance of well-posedness: In practice, the initial and boundary data are measured and so small errors occur. Very often the problem must be solved numerically which involves truncation and round-off errors. If the problem is well-posed then these unavoidable small errors produce only slight errors in the computed solution, and, hence, useful results are obtained.

  6. Introduction Classifications Canonical forms Separation of variables Outline Introduction 1 Basic notions and notations Methods and techniques for solving PDEs Well-posed and ill-posed problems Classifications 2 Basic classifications of PDEs Kinds of nonlinearity Types of second-order linear PDEs Classic linear PDEs 3 Canonical forms Canonical forms of second order PDEs Reduction to a canonical form Transforming the hyperbolic equation Separation of variables 4 Necessary assumptions Explanation of the method

  7. Introduction Classifications Canonical forms Separation of variables Basic classifications of PDEs Order of the PDE. The order of a PDE is the order of the highest partial derivative in the equation. Example first order: u t = u x , second order: u t = u xx , u xy = 0 , third order: u t + u u xxx = sin( x ) fourth order: u xxxx = u tt .

  8. Introduction Classifications Canonical forms Separation of variables Basic classifications of PDEs Order of the PDE. The order of a PDE is the order of the highest partial derivative in the equation. Number of variables. PDEs may be classified by the number of their independent variables, that is, the number of variables the unknown function depends on. Example � � PDE in two variables: u t = u xx , u = u ( t , x ) , � � u t = u rr + 1 r u r + 1 PDE in three variables: u = u ( t , r , θ ) r 2 u θθ , , � � u t = u xx + u yy + u zz , u = u ( t , x , y , z ) PDE in four variables: .

  9. Introduction Classifications Canonical forms Separation of variables Basic classifications of PDEs Order of the PDE. The order of a PDE is the order of the highest partial derivative in the equation. Number of variables. PDEs may be classified by the number of their independent variables, that is, the number of variables the unknown function depends on. Linearity. PDE is linear if the dependent variable and all its derivatives appear in a linear fashion. Example linear: u tt + exp( − t ) u xx = sin( t ) , nonlinear: u u xx + u t = 0 , linear: x u xx + y u yy = 0 , u x + u y + u 2 = 0 . nonlinear:

  10. Introduction Classifications Canonical forms Separation of variables Basic classifications of PDEs Order of the PDE. The order of a PDE is the order of the highest partial derivative in the equation. Number of variables. PDEs may be classified by the number of their independent variables, that is, the number of variables the unknown function depends on. Linearity. PDE is linear if the dependent variable and all its derivatives appear in a linear fashion. Kinds of coefficients. PDE can be with constant or variable coefficients (if at least one of the coefficients is a function of (some of) independent variables). Example constant coefficients: u tt + 5 u xx − 3 u xy = cos( x ) , variable coefficients: u t + exp( − t ) u xx = 0 .

  11. Introduction Classifications Canonical forms Separation of variables Basic classifications of PDEs Order of the PDE. The order of a PDE is the order of the highest partial derivative in the equation. Number of variables. PDEs may be classified by the number of their independent variables, that is, the number of variables the unknown function depends on. Linearity. PDE is linear if the dependent variable and all its derivatives appear in a linear fashion. Kinds of coefficients. PDE can be with constant or variable coefficients (if at least one of the coefficients is a function of (some of) independent variables). Homogeneity. PDE is homogeneous if the free term (the right-hand side term) is zero. Example homogeneous: u tt − u xx = 0 , u tt − u xx = x 2 sin( t ) . nonhomogeneous:

  12. Introduction Classifications Canonical forms Separation of variables Basic classifications of PDEs Order of the PDE. The order of a PDE is the order of the highest partial derivative in the equation. Number of variables. PDEs may be classified by the number of their independent variables, that is, the number of variables the unknown function depends on. Linearity. PDE is linear if the dependent variable and all its derivatives appear in a linear fashion. Kinds of coefficients. PDE can be with constant or variable coefficients (if at least one of the coefficients is a function of (some of) independent variables). Homogeneity. PDE is homogeneous if the free term (the right-hand side term) is zero. Kind of PDE. All linear second-order PDEs are either: hyperbolic (e.g., u tt − u xx = f ( t , x , u , u t , u x ) ), parabolic (e.g., u xx = f ( t , x , u , u t , u x ) ), elliptic (e.g., u xx + u yy = f ( x , y , u , u x , u y ) ).

  13. Introduction Classifications Canonical forms Separation of variables Kinds of nonlinearity Definition (Semi-linearity, quasi-linearity, and full nonlinearity) A partial differential equation is: semi-linear – if the highest derivatives appear in a linear fashion and their coefficients do not depend on the unknown function or its derivatives; quasi-linear – if the highest derivatives appear in a linear fashion; fully nonlinear – if the highest derivatives appear in a nonlinear fashion.

  14. Introduction Classifications Canonical forms Separation of variables Kinds of nonlinearity Definition (Semi-linearity, quasi-linearity, and full nonlinearity) A partial differential equation is: semi-linear – if the highest derivatives appear in a linear fashion and their coefficients do not depend on the unknown function or its derivatives; quasi-linear – if the highest derivatives appear in a linear fashion; fully nonlinear – if the highest derivatives appear in a nonlinear fashion. Let: u = u ( x ) and x = ( x , y ) . Example (semi-linear PDE) C 1 ( x ) u xx + C 2 ( x ) u xy + C 3 ( x ) u yy + C 0 ( x , u , u x , u y ) = 0

  15. Introduction Classifications Canonical forms Separation of variables Kinds of nonlinearity Definition (Semi-linearity, quasi-linearity, and full nonlinearity) A partial differential equation is: semi-linear – if the highest derivatives appear in a linear fashion and their coefficients do not depend on the unknown function or its derivatives; quasi-linear – if the highest derivatives appear in a linear fashion; fully nonlinear – if the highest derivatives appear in a nonlinear fashion. Let: u = u ( x ) and x = ( x , y ) . Example (quasi-linear PDE) C 1 ( x , u , u x , u y ) u xx + C 2 ( x , u , u x , u y ) u xy + C 0 ( x , u , u x , u y ) = 0

  16. Introduction Classifications Canonical forms Separation of variables Kinds of nonlinearity Definition (Semi-linearity, quasi-linearity, and full nonlinearity) A partial differential equation is: semi-linear – if the highest derivatives appear in a linear fashion and their coefficients do not depend on the unknown function or its derivatives; quasi-linear – if the highest derivatives appear in a linear fashion; fully nonlinear – if the highest derivatives appear in a nonlinear fashion. Let: u = u ( x ) and x = ( x , y ) . Example (fully non-linear PDE) u xx u xy = 0

  17. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A second-order linear PDE in two variables can be in general written in the following form A u xx + B u xy + C u yy + D u x + E u y + F u = G where A , B , C , D , E , and F are coefficients, and G is a right-hand side (i.e., non-homogeneous) term. All these quantities are constants, or at most, functions of ( x , y ) .

  18. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0

  19. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0 Example B 2 − 4 AC = 0 2 − 4 · ( − 1 ) · 1 = 4 > 0 , u tt − u xx = 0 → B 2 − 4 AC = 1 2 − 4 · 0 · 0 = 1 > 0 . u tx = 0 →

  20. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0 (eg., u tt − u xx = 0 , u tx = 0 ), parabolic: if B 2 − 4 AC = 0

  21. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0 (eg., u tt − u xx = 0 , u tx = 0 ), parabolic: if B 2 − 4 AC = 0 Example B 2 − 4 AC = 0 2 − 4 · ( − 1 ) · 0 = 0 . u t − u xx = 0 →

  22. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0 (eg., u tt − u xx = 0 , u tx = 0 ), parabolic: if B 2 − 4 AC = 0 (eg., u t − u xx = 0 ), elliptic: if B 2 − 4 AC < 0

  23. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0 (eg., u tt − u xx = 0 , u tx = 0 ), parabolic: if B 2 − 4 AC = 0 (eg., u t − u xx = 0 ), elliptic: if B 2 − 4 AC < 0 Example B 2 − 4 AC = 0 2 − 4 · 1 · 1 = − 4 < 0 . u xx + u yy = 0 →

  24. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0 (eg., u tt − u xx = 0 , u tx = 0 ), parabolic: if B 2 − 4 AC = 0 (eg., u t − u xx = 0 ), elliptic: if B 2 − 4 AC < 0 (eg., u xx + u yy = 0 ). The mathematical solutions to these three types of equations are quite different. The three major classifications of linear PDEs essentially classify physical problems into three basic types: 1 vibrating systems and wave propagation ( hyperbolic case), 2 heat flow and diffusion processes ( parabolic case), 3 steady-state phenomena ( elliptic case).

  25. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0 (eg., u tt − u xx = 0 , u tx = 0 ), parabolic: if B 2 − 4 AC = 0 (eg., u t − u xx = 0 ), elliptic: if B 2 − 4 AC < 0 (eg., u xx + u yy = 0 ). In general, ( B 2 − 4 A C ) is a function of the independent variables ( x , y ) . Hence, an equation can change from one basic type to another. Example    > 0 for y < 0 (hyperbolic),  B 2 − 4 AC = − 4 y y u xx + u yy = 0 → = 0 for y = 0 (parabolic),    < 0 for y > 0 (elliptic).

  26. Introduction Classifications Canonical forms Separation of variables Types of second-order linear PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G The second-order linear PDE is either hyperbolic: if B 2 − 4 AC > 0 (eg., u tt − u xx = 0 , u tx = 0 ), parabolic: if B 2 − 4 AC = 0 (eg., u t − u xx = 0 ), elliptic: if B 2 − 4 AC < 0 (eg., u xx + u yy = 0 ). Second-order linear equations in three or more variables can also be classified except that matrix analysis must be used. Example u t = u xx + u yy ← parabolic equation , u tt = u xx + u yy + u zz ← hyperbolic equation .

  27. Introduction Classifications Canonical forms Separation of variables Classic linear PDEs Hyperbolic PDEs: u tt − c 2 u xx = 0 Vibrating string (1D wave equation): u tt − c 2 ∇ 2 u + h u t = 0 Wave equation with damping (if h � = 0 ): u tt − c 2 ∇ 2 u + h u t + k u = 0 Transmission line equation: Parabolic PDEs: u t − α 2 u xx + h u x = 0 Diffusion-convection equation: Diffusion with lateral heat-concentration loss: u t − α 2 u xx + k u = 0 Elliptic PDEs: ∇ 2 u = 0 Laplace’s equation: ∇ 2 u = k Poisson’s equation: ∇ 2 u + λ 2 u = 0 Helmholtz’s equation: ∇ 2 u + k ( E − V ) u = 0 Shr¨ odinger’s equation: Higher-order PDEs: Airy’s equation (third order): u t + u xxx = 0 α 2 u tt + u xxxx = 0 Bernouli’s beam equation (fourth order): α 2 u tt + ∇ 4 u = 0 Kirchhoff’s plate equation (fourth order): (Here: ∇ 2 is the Laplace operator, ∇ 4 = ∇ 2 ∇ 2 is the biharmonic operator.)

  28. Introduction Classifications Canonical forms Separation of variables Outline Introduction 1 Basic notions and notations Methods and techniques for solving PDEs Well-posed and ill-posed problems Classifications 2 Basic classifications of PDEs Kinds of nonlinearity Types of second-order linear PDEs Classic linear PDEs 3 Canonical forms Canonical forms of second order PDEs Reduction to a canonical form Transforming the hyperbolic equation Separation of variables 4 Necessary assumptions Explanation of the method

  29. Introduction Classifications Canonical forms Separation of variables Canonical forms of second order PDEs Any second-order linear PDE (in two variables) A u xx + B u xy + C u yy + D u x + E u y + F u = G (where A , B , C , D , E , F , and G are constants or functions of ( x , y ) ) can be transformed into the so-called canonical form . This can be achieved by introducing new coordinates : ξ = ξ ( x , y ) and η = η ( x , y ) (in place of x , y ) which simplify the equation to its canonical form.

  30. Introduction Classifications Canonical forms Separation of variables Canonical forms of second order PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G can be transformed into its canonical form by introducing new coordinates : ξ = ξ ( x , y ) and η = η ( x , y ) The type of PDE determines the canonical form: ◮ for hyperbolic PDE (that is, when B 2 − 4 A C > 0 ) there are, in fact, two possibilities: � ˜ � B 2 − 4 ˜ C = 0 2 − 4 · 1 · ( − 1 ) = 4 > 0 A ˜ u ξξ − u ηη = f ( ξ, η, u , u ξ , u η ) , � ˜ � B 2 − 4 ˜ C = 1 2 − 4 · 0 · 0 = 1 > 0 A ˜ or u ξη = f ( ξ, η, u , u ξ , u η ) ;

  31. Introduction Classifications Canonical forms Separation of variables Canonical forms of second order PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G can be transformed into its canonical form by introducing new coordinates : ξ = ξ ( x , y ) and η = η ( x , y ) The type of PDE determines the canonical form: ◮ for hyperbolic PDE (that is, when B 2 − 4 A C > 0 ) there are, in fact, two possibilities: � ˜ � B 2 − 4 ˜ C = 0 2 − 4 · 1 · ( − 1 ) = 4 > 0 A ˜ u ξξ − u ηη = f ( ξ, η, u , u ξ , u η ) , � ˜ � B 2 − 4 ˜ C = 1 2 − 4 · 0 · 0 = 1 > 0 A ˜ or u ξη = f ( ξ, η, u , u ξ , u η ) ; ◮ for parabolic PDE (that is, when B 2 − 4 A C = 0 ): � ˜ � B 2 − 4 ˜ C = 0 2 − 4 · 1 · 0 = 0 A ˜ u ξξ = f ( ξ, η, u , u ξ , u η ) ;

  32. Introduction Classifications Canonical forms Separation of variables Canonical forms of second order PDEs A u xx + B u xy + C u yy + D u x + E u y + F u = G can be transformed into its canonical form by introducing new coordinates : ξ = ξ ( x , y ) and η = η ( x , y ) The type of PDE determines the canonical form: ◮ for hyperbolic PDE (that is, when B 2 − 4 A C > 0 ) there are, in fact, two possibilities: � ˜ � B 2 − 4 ˜ C = 0 2 − 4 · 1 · ( − 1 ) = 4 > 0 A ˜ u ξξ − u ηη = f ( ξ, η, u , u ξ , u η ) , � ˜ � B 2 − 4 ˜ C = 1 2 − 4 · 0 · 0 = 1 > 0 A ˜ or u ξη = f ( ξ, η, u , u ξ , u η ) ; ◮ for parabolic PDE (that is, when B 2 − 4 A C = 0 ): � ˜ � B 2 − 4 ˜ C = 0 2 − 4 · 1 · 0 = 0 A ˜ u ξξ = f ( ξ, η, u , u ξ , u η ) ; ◮ for elliptic PDE (that is, when B 2 − 4 A C < 0 ): � ˜ � B 2 − 4 ˜ C = 0 2 − 4 · 1 · 1 = − 4 < 0 A ˜ u ξξ + u ηη = f ( ξ, η, u , u ξ , u η ) .

  33. Introduction Classifications Canonical forms Separation of variables Reduction to a canonical form Step 1. Introduce new coordinates ξ = ξ ( x , y ) and η = η ( x , y ) .

  34. Introduction Classifications Canonical forms Separation of variables Reduction to a canonical form Step 1. Introduce new coordinates ξ = ξ ( x , y ) and η = η ( x , y ) . Compute the partial derivatives: u x = u ξ ξ x + u η η x , u y = u ξ ξ y + u η η y , u xx = u ξξ ξ 2 x + 2 u ξη ξ x η x + u ηη η 2 x + u ξ ξ xx + u η η xx , u yy = u ξξ ξ 2 y + 2 u ξη ξ y η y + u ηη η 2 y + u ξ ξ yy + u η η yy , � � u xy = u ξξ ξ x ξ y + u ξη ξ x η y + ξ y η x + u ηη η x η y + u ξ ξ xy + u η η xy .

  35. Introduction Classifications Canonical forms Separation of variables Reduction to a canonical form Step 1. Introduce new coordinates ξ = ξ ( x , y ) and η = η ( x , y ) . Compute the partial derivatives: u x = u ξ ξ x + u η η x , u y = u ξ ξ y + u η η y , u xx = u ξξ ξ 2 x + 2 u ξη ξ x η x + u ηη η 2 x + u ξ ξ xx + u η η xx , u yy = u ξξ ξ 2 y + 2 u ξη ξ y η y + u ηη η 2 y + u ξ ξ yy + u η η yy , � � u xy = u ξξ ξ x ξ y + u ξη ξ x η y + ξ y η x + u ηη η x η y + u ξ ξ xy + u η η xy . Substitute these values into the original equation to obtain a new form: A u ξξ + � � B u ξη + � C u ηη + � D u ξ + � E u η + F u = G where the new coefficients are as follows � � A = A ξ 2 � x + B ξ x ξ y + C ξ 2 � B = 2 A ξ x η x + B ξ x η y + ξ y η x + 2 C ξ y η y , y , C = A η 2 � x + B η x η y + C η 2 � y , D = A ξ xx + B ξ xy + C ξ yy + D ξ x + E ξ y , � E = A η xx + B η xy + C η yy + D η x + E η y .

  36. Introduction Classifications Canonical forms Separation of variables Reduction to a canonical form Step 1. Introduce new coordinates ξ = ξ ( x , y ) and η = η ( x , y ) . � A u ξξ + � B u ξη + � C u ηη + � D u ξ + � E u η + F u = G Step 2. Impose the requirements onto coefficients � B , � A , � C , and solve for ξ and η .

  37. Introduction Classifications Canonical forms Separation of variables Reduction to a canonical form Step 1. Introduce new coordinates ξ = ξ ( x , y ) and η = η ( x , y ) . � A u ξξ + � B u ξη + � C u ηη + � D u ξ + � E u η + F u = G Step 2. Impose the requirements onto coefficients � B , � A , � C , and solve for ξ and η . The requirements depend on the type of the PDE, namely: C = 0 for the hyperbolic PDE (when B 2 − 4 A C > 0 ); set � A = �

  38. Introduction Classifications Canonical forms Separation of variables Reduction to a canonical form Step 1. Introduce new coordinates ξ = ξ ( x , y ) and η = η ( x , y ) . � A u ξξ + � B u ξη + � C u ηη + � D u ξ + � E u η + F u = G Step 2. Impose the requirements onto coefficients � B , � A , � C , and solve for ξ and η . The requirements depend on the type of the PDE, namely: C = 0 for the hyperbolic PDE (when B 2 − 4 A C > 0 ); set � A = � set either � A = 0 or � C = 0 for the parabolic PDE; in this case another necessary requirement � B = 0 will follow automatically (since B 2 − 4 A C = 0 );

  39. Introduction Classifications Canonical forms Separation of variables Reduction to a canonical form Step 1. Introduce new coordinates ξ = ξ ( x , y ) and η = η ( x , y ) . � A u ξξ + � B u ξη + � C u ηη + � D u ξ + � E u η + F u = G Step 2. Impose the requirements onto coefficients � B , � A , � C , and solve for ξ and η . The requirements depend on the type of the PDE, namely: C = 0 for the hyperbolic PDE (when B 2 − 4 A C > 0 ); set � A = � set either � A = 0 or � C = 0 for the parabolic PDE; in this case another necessary requirement � B = 0 will follow automatically (since B 2 − 4 A C = 0 ); for the elliptic PDE (when B 2 − 4 A C < 0 ), firstly, proceed as in the hyperbolic case: set � A = � C = 0 to find the complex conjugate coordinates ξ , η (which would lead to a form of complex hyperbolic equation u ξη = f ( ξ, η, u , u ξ , u η ) ); then, transform ξ and η as follows: α ← ξ + η β ← ξ − η , . 2 2 i (Here, α is the real part of ξ and η , while β is the imaginary part.) The new real coordinates, α and β , allow to write the final canonical elliptic form: u αα + u ββ = f ( α, β, u , u α , u β ) .

  40. Introduction Classifications Canonical forms Separation of variables Reduction to a canonical form Step 1. Introduce new coordinates ξ = ξ ( x , y ) and η = η ( x , y ) . � A u ξξ + � B u ξη + � C u ηη + � D u ξ + � E u η + F u = G Step 2. Impose the requirements onto coefficients � B , � A , � C , and solve for ξ and η . Step 3. Use the new coordinates for the coefficients and homogeneous term of the new canonical form (i.e., replace x = x ( ξ, η ) and y = y ( ξ, η ) ).

  41. Introduction Classifications Canonical forms Separation of variables Transforming the hyperbolic equation For hyperbolic equation the canonical form u ξη = f ( ξ, η, u , u ξ , u η ) ✞ ☎ A = � � is achieved by setting C = 0 , ✝ ✆

  42. Introduction Classifications Canonical forms Separation of variables Transforming the hyperbolic equation For hyperbolic equation the canonical form u ξη = f ( ξ, η, u , u ξ , u η ) ✞ ☎ A = � � is achieved by setting C = 0 , that is, ✝ ✆ � � A = A ξ 2 x + B ξ x ξ y + C ξ 2 C = A η 2 x + B η x η y + C η 2 y = 0 , y = 0 , which can be rewritten as � ξ x � 2 � η x � 2 + B ξ x + B η x A + C = 0 , A + C = 0 . ξ y ξ y η y η y

  43. Introduction Classifications Canonical forms Separation of variables Transforming the hyperbolic equation For hyperbolic equation the canonical form u ξη = f ( ξ, η, u , u ξ , u η ) ✞ ☎ A = � � is achieved by setting C = 0 , that is, ✝ ✆ � � A = A ξ 2 x + B ξ x ξ y + C ξ 2 C = A η 2 x + B η x η y + C η 2 y = 0 , y = 0 , which can be rewritten as � ξ x � 2 � η x � 2 + B ξ x + B η x A + C = 0 , A + C = 0 . ξ y ξ y η y η y Solving these equations for ξ x ξ y and η x η y one finds the so-called characteristic equations : √ √ B 2 − 4 A C B 2 − 4 A C ξ x = − B + η x = − B − , . ξ y 2 A η y 2 A

  44. Introduction Classifications Canonical forms Separation of variables Transforming the hyperbolic equation The new coordinates equated to constant values define the parametric lines of the new system of coordinates. That means that the total derivatives are zero, i.e., d y d x = − ξ x ξ ( x , y ) = const . → d ξ = ξ x d x + ξ y d y = 0 → , ξ y d y d x = − η x η ( x , y ) = const . d η = η x d x + η y d y = 0 → → , η y

  45. Introduction Classifications Canonical forms Separation of variables Transforming the hyperbolic equation The new coordinates equated to constant values define the parametric lines of the new system of coordinates. That means that the total derivatives are zero, i.e., d y d x = − ξ x ξ ( x , y ) = const . → d ξ = ξ x d x + ξ y d y = 0 → , ξ y d y d x = − η x η ( x , y ) = const . d η = η x d x + η y d y = 0 → → , η y Therefore, the characteristic equations are √ √ B 2 − 4 A C B 2 − 4 A C d y d x = − ξ x = B − d x = − η x d y = B + , , ξ y 2 A η y 2 A and can be easily integrated to find the implicit solutions, ξ ( x , y ) = const . and η ( x , y ) = const . , that is, the new coordinates ensuring the simple canonical form of the PDE.

  46. Introduction Classifications Canonical forms Separation of variables Example Rewriting a hyperbolic equation in canonical form y 2 u xx − x 2 u yy = 0 x ∈ ( 0 , + ∞ ) , y ∈ ( 0 , + ∞ ) . (In the first quadrant this is a hyperbolic equation, since B 2 − 4 A C = 4 y 2 x 2 > 0 for x � = 0 and y � = 0 .)

  47. Introduction Classifications Canonical forms Separation of variables Example Rewriting a hyperbolic equation in canonical form y 2 u xx − x 2 u yy = 0 x ∈ ( 0 , + ∞ ) , y ∈ ( 0 , + ∞ ) . Writing the two characteristic equations √ √ B 2 − 4 A C B 2 − 4 A C d x = B − d y = − x d y d x = B + = x y , y . 2 A 2 A

  48. Introduction Classifications Canonical forms Separation of variables Example Rewriting a hyperbolic equation in canonical form y 2 u xx − x 2 u yy = 0 x ∈ ( 0 , + ∞ ) , y ∈ ( 0 , + ∞ ) . Writing the two characteristic equations √ √ B 2 − 4 A C B 2 − 4 A C d y d x = B − = − x d y d x = B + = x y , y . 2 A 2 A Solving these equations – by separating the variables y d y = − x d x , y d y = x d x , and integrating ξ ( x , y ) = y 2 + x 2 = const . , η ( x , y ) = y 2 − x 2 = const .

  49. Introduction Classifications Canonical forms Separation of variables Example Rewriting a hyperbolic equation in canonical form y 2 u xx − x 2 u yy = 0 x ∈ ( 0 , + ∞ ) , y ∈ ( 0 , + ∞ ) . Writing the two characteristic equations √ √ B 2 − 4 A C B 2 − 4 A C d y d x = B − = − x d y d x = B + = x y , y . 2 A 2 A Solving these equations – by separating the variables and integrating y d y = − x d x , y d y = x d x , ξ ( x , y ) = y 2 + x 2 = const . , η ( x , y ) = y 2 − x 2 = const . Using the new coordinates for the (non-zero) coefficients B = − 16 x 2 y 2 = 4 ( η 2 − ξ 2 ) , � D = − 2 ( y 2 + x 2 ) = − 2 ξ , � � E = 2 ( y 2 − x 2 ) = 2 η , to present the PDE in the canonical form : D u ξ + � � E u η = ξ u ξ − η u η u ξη = 2 ( ξ 2 − η 2 ) . � B

  50. Introduction Classifications Canonical forms Separation of variables Example New coordinates for the canonical form of the hyperbolic PDE y η = 0 16 9 4 1 5 − 1 PDE in ( x , y ) : y 2 u xx − x 2 u yy = 0 − 4 − 9 4 PDE in ( ξ, η ) : u ξη = ξ u ξ − η u η 2 ( ξ 2 − η 2 ) 3 − 16 η ( x , y ) = const . ← hyperbolas 2 ξ ( x , y ) = const. ← circles ξ = 1 x 0 1 2 3 4 5 ξ ( x , y ) = y 2 + x 2 = const . ∈ ( 0 , + ∞ ) , η ( x , y ) = y 2 − x 2 = const . ∈ ( −∞ , + ∞ ) .

  51. Introduction Classifications Canonical forms Separation of variables Outline Introduction 1 Basic notions and notations Methods and techniques for solving PDEs Well-posed and ill-posed problems Classifications 2 Basic classifications of PDEs Kinds of nonlinearity Types of second-order linear PDEs Classic linear PDEs 3 Canonical forms Canonical forms of second order PDEs Reduction to a canonical form Transforming the hyperbolic equation Separation of variables 4 Necessary assumptions Explanation of the method

  52. Introduction Classifications Canonical forms Separation of variables Separation of variables Necessary assumptions This technique applies to problems which satisfy two requirements. 1 The PDE is linear and homogeneous (not necessary constant coefficients). 2 The boundary conditions are linear and homogeneous .

  53. Introduction Classifications Canonical forms Separation of variables Separation of variables Necessary assumptions This technique applies to problems which satisfy two requirements. 1 The PDE is linear and homogeneous . A second-order PDE in two variables ( x and t ) is linear and homogeneous , if it can be written in the following form A u xx + B u xt + C u tt + D u x + E u t + F u = 0 where the coefficients A , B , C , D , E , and F do not depend on the dependent variable u = u ( x , t ) or any of its derivatives though can be functions of independent variables ( x , t ) . The boundary conditions are linear and homogeneous . 2

  54. Introduction Classifications Canonical forms Separation of variables Separation of variables Necessary assumptions This technique applies to problems which satisfy two requirements. 1 The PDE is linear and homogeneous . A second-order PDE in two variables ( x and t ) is linear and homogeneous , if it can be written in the following form A u xx + B u xt + C u tt + D u x + E u t + F u = 0 where the coefficients A , B , C , D , E , and F do not depend on the dependent variable u = u ( x , t ) or any of its derivatives though can be functions of independent variables ( x , t ) . The boundary conditions are linear and homogeneous . 2 In the case of the second-order PDE, a general form of such boundary conditions is G 1 u x ( x 1 , t ) + H 1 u ( x 1 , t ) = 0 , G 2 u x ( x 2 , t ) + H 2 u ( x 2 , t ) = 0 , where G 1 , G 2 , H 1 , H 2 are constants.

  55. Introduction Classifications Canonical forms Separation of variables Separation of variables Scheme of the method Main procedure : 1 break down the initial conditions into simple components,

  56. Introduction Classifications Canonical forms Separation of variables Separation of variables Scheme of the method Main procedure : 1 break down the initial conditions into simple components, 2 find the response to each component,

  57. Introduction Classifications Canonical forms Separation of variables Separation of variables Scheme of the method Main procedure : 1 break down the initial conditions into simple components, 2 find the response to each component, 3 add up these individual responses to obtain the final result.

  58. Introduction Classifications Canonical forms Separation of variables Separation of variables Scheme of the method Main procedure : 1 break down the initial conditions into simple components, 2 find the response to each component, 3 add up these individual responses to obtain the final result. The separation of variables technique looks first for the so-called fundamental solutions . They are simple-type solutions of the form u i ( x , t ) = X i ( x ) T i ( t ) , where X i ( x ) is a sort of “shape” of the solution i whereas T i ( t ) scales this “shape” for different values of time t .

  59. Introduction Classifications Canonical forms Separation of variables Separation of variables Scheme of the method Main procedure : 1 break down the initial conditions into simple components, 2 find the response to each component, 3 add up these individual responses to obtain the final result. The separation of variables technique looks first for the so-called fundamental solutions . They are simple-type solutions of the form u i ( x , t ) = X i ( x ) T i ( t ) , where X i ( x ) is a sort of “shape” of the solution i whereas T i ( t ) scales this “shape” for different values of time t . The fundamental solution will : always retain its basic “shape” , at the same time, satisfy the BCs which puts a requirement only on the “shape” function X i ( x ) since the BCs are linear and homogeneous. The general idea is that it is possible to find an infinite number of these fundamental solutions (everyone corresponding to an adequate simple component of initial conditions).

  60. Introduction Classifications Canonical forms Separation of variables Separation of variables Scheme of the method Main procedure : 1 break down the initial conditions into simple components, 2 find the response to each component, 3 add up these individual responses to obtain the final result. The separation of variables technique looks first for the so-called fundamental solutions . They are simple-type solutions of the form u i ( x , t ) = X i ( x ) T i ( t ) , where X i ( x ) is a sort of “shape” of the solution i whereas T i ( t ) scales this “shape” for different values of time t . The solution of the problem is found by adding the simple fundamental solutions in such a way that the resulting sum � n � n u ( x , t ) = a i u i ( x , t ) = a i X i ( x ) T i ( t ) i = 1 i = 1 satisfies the initial conditions which is attained by a proper selection of the coefficients a i .

  61. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method IBVP for heat flow (or diffusion process) Find u = u ( x , t ) =? satisfying for x ∈ [ 0 , 1 ] and t ∈ [ 0 , ∞ ) : � u ( 0 , t ) = 0 , PDE: u t = α 2 u xx , BCs: IC: u ( x , 0 ) = f ( x ) , u x ( 1 , t ) + h u ( 1 , t ) = 0 , where α , h , and f ( x ) are some known constants or functions.

  62. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method IBVP for heat flow (or diffusion process) Find u = u ( x , t ) =? satisfying for x ∈ [ 0 , 1 ] and t ∈ [ 0 , ∞ ) : � u ( 0 , t ) = 0 , PDE: u t = α 2 u xx , BCs: IC: u ( x , 0 ) = f ( x ) , u x ( 1 , t ) + h u ( 1 , t ) = 0 , where α , h , and f ( x ) are some known constants or functions. Step 1. Separating the PDE into two ODEs. ◮ Substituting the separated form (of the fundamental solution), u ( x , t ) = u i ( x , t ) = X i ( x ) T i ( t ) , into the PDE gives (after division by α 2 X i ( x ) T i ( t ) ) α 2 T i ( t ) = X ′′ T ′ i ( t ) i ( x ) X i ( x ) . ◮ Both sides of this equation must be constant (since they depend only on x or t which are independent ). Setting them both equal to µ i results in two ODEs: i ( t ) − µ i α 2 T i ( t ) = 0 , T ′ X ′′ i ( x ) − µ i X i ( x ) = 0 .

  63. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method IBVP for heat flow (or diffusion process) Find u = u ( x , t ) =? satisfying for x ∈ [ 0 , 1 ] and t ∈ [ 0 , ∞ ) : � u ( 0 , t ) = 0 , PDE: u t = α 2 u xx , BCs: IC: u ( x , 0 ) = f ( x ) , u x ( 1 , t ) + h u ( 1 , t ) = 0 , where α , h , and f ( x ) are some known constants or functions. Step 1. Separating the PDE into two ODEs. Step 2. Finding the separation constant and fundamental solutions. If µ i = 0 then: (after using the BCs) a trivial solution u ( x , t ) ≡ 0 is obtained. For µ i > 0 : T ( t ) (and so u ( x , t ) = X ( x ) T ( t ) ) will grow exponentially to infinity which can be rejected on physical grounds. Therefore: µ i = − λ 2 i < 0 .

  64. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method IBVP for heat flow (or diffusion process) Find u = u ( x , t ) =? satisfying for x ∈ [ 0 , 1 ] and t ∈ [ 0 , ∞ ) : � u ( 0 , t ) = 0 , PDE: u t = α 2 u xx , BCs: IC: u ( x , 0 ) = f ( x ) , u x ( 1 , t ) + h u ( 1 , t ) = 0 , where α , h , and f ( x ) are some known constants or functions. Step 1. Separating the PDE into two ODEs. Step 2. Finding the separation constant and fundamental solutions. ◮ Now, the two ODEs can be written as i α 2 T i ( t ) = 0 , T ′ i ( t ) + λ 2 X ′′ i ( x ) + λ 2 i X i ( x ) = 0 , and solutions to them are � � i α 2 t T i ( t ) = ˜ − λ 2 X i ( x ) = ˜ C 1 sin( λ i x ) + ˜ C 0 exp , C 2 cos( λ i x ) , where ˜ C 0 , ˜ C 1 , and ˜ C 2 are constants. ◮ That leads to the following fundamental solution (with constants C 1 , C 2 ) � � i α 2 t ) . exp( − λ 2 u i ( x , t ) = X i ( x ) T i ( t ) = C 1 sin( λ i x ) + C 2 cos( λ i x )

  65. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method IBVP for heat flow (or diffusion process) Find u = u ( x , t ) =? satisfying for x ∈ [ 0 , 1 ] and t ∈ [ 0 , ∞ ) : � u ( 0 , t ) = 0 , PDE: u t = α 2 u xx , BCs: IC: u ( x , 0 ) = f ( x ) , u x ( 1 , t ) + h u ( 1 , t ) = 0 , where α , h , and f ( x ) are some known constants or functions. Step 1. Separating the PDE into two ODEs. Step 2. Finding the separation constant and fundamental solutions. ◮ Applying the boundary conditions i α 2 t ) = 0 C 2 exp( − λ 2 at x = 0 : C 2 = 0 , → � � tan λ i = − λ i i α 2 t ) C 1 exp( − λ 2 at x = 1 : λ i cos( λ i ) + h sin( λ i ) = 0 → h . SOLVE (they are eigenvalues for That gives a desired condition on λ i which there exists a nonzero solution). ◮ The fundamental solutions are as follows PLOT i α 2 t ) . u i ( x , t ) = sin( λ i x ) exp( − λ 2

  66. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method IBVP for heat flow (or diffusion process) Find u = u ( x , t ) =? satisfying for x ∈ [ 0 , 1 ] and t ∈ [ 0 , ∞ ) : � u ( 0 , t ) = 0 , PDE: u t = α 2 u xx , BCs: IC: u ( x , 0 ) = f ( x ) , u x ( 1 , t ) + h u ( 1 , t ) = 0 , where α , h , and f ( x ) are some known constants or functions. Step 1. Separating the PDE into two ODEs. Step 2. Finding the separation constant and fundamental solutions. Step 3. Expansion of the IC as a sum of eigenfunctions. ◮ The final solution is such linear combination (with coefficients a i ) of infinite number of fundamental solutions, � � ∞ ∞ i α 2 t ) , a i sin( λ i x ) exp( − λ 2 u ( x , t ) = a i u i ( x , t ) = i = 1 i = 1 that satisfies the initial condition: � ∞ f ( x ) ≡ u ( x , 0 ) = a i sin( λ i x ) . i = 1

  67. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method Step 1. Separating the PDE into two ODEs. Step 2. Finding the separation constant and fundamental solutions. Step 3. Expansion of the IC as a sum of eigenfunctions. ◮ The final solution is such linear combination (with coefficients a i ) of infinite number of fundamental solutions, � ∞ � ∞ i α 2 t ) , a i sin( λ i x ) exp( − λ 2 u ( x , t ) = a i u i ( x , t ) = i = 1 i = 1 that satisfies the initial condition: � ∞ f ( x ) ≡ u ( x , 0 ) = a i sin( λ i x ) . i = 1 ◮ The coefficients a i in the eigenfunction expansion are found by multiplying both sides of the IC equation by sin( λ j x ) and integrating using the orthogonality property, i.e., � � 1 1 � ∞ f ( x ) sin( λ j x ) d x = sin( λ i x ) sin( λ j x ) d x a i i = 1 0 0

  68. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method Step 1. Separating the PDE into two ODEs. Step 2. Finding the separation constant and fundamental solutions. Step 3. Expansion of the IC as a sum of eigenfunctions. ◮ The final solution is such linear combination (with coefficients a i ) of infinite number of fundamental solutions, � ∞ � ∞ i α 2 t ) , a i sin( λ i x ) exp( − λ 2 u ( x , t ) = a i u i ( x , t ) = i = 1 i = 1 that satisfies the initial condition: � ∞ f ( x ) ≡ u ( x , 0 ) = a i sin( λ i x ) . i = 1 ◮ The coefficients a i in the eigenfunction expansion are found by multiplying both sides of the IC equation by sin( λ j x ) and integrating using the orthogonality property, i.e., � � 1 1 sin 2 ( λ j x ) d x f ( x ) sin( λ j x ) d x = a j 0 0

  69. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method Step 1. Separating the PDE into two ODEs. Step 2. Finding the separation constant and fundamental solutions. Step 3. Expansion of the IC as a sum of eigenfunctions. ◮ The final solution is such linear combination (with coefficients a i ) of infinite number of fundamental solutions, � ∞ � ∞ i α 2 t ) , a i sin( λ i x ) exp( − λ 2 u ( x , t ) = a i u i ( x , t ) = i = 1 i = 1 that satisfies the initial condition: � ∞ f ( x ) ≡ u ( x , 0 ) = a i sin( λ i x ) . i = 1 ◮ The coefficients a i in the eigenfunction expansion are found by multiplying both sides of the IC equation by sin( λ j x ) and integrating using the orthogonality property, i.e., � 1 f ( x ) sin( λ j x ) d x = a j λ j − sin( λ j ) cos( λ j ) 2 λ j 0

  70. Introduction Classifications Canonical forms Separation of variables Example Solving a parabolic IBVP by the separation of variables method Step 1. Separating the PDE into two ODEs. Step 2. Finding the separation constant and fundamental solutions. Step 3. Expansion of the IC as a sum of eigenfunctions. ◮ The final solution is such linear combination (with coefficients a i ) of infinite number of fundamental solutions, � ∞ � ∞ i α 2 t ) , a i sin( λ i x ) exp( − λ 2 u ( x , t ) = a i u i ( x , t ) = i = 1 i = 1 that satisfies the initial condition: � ∞ f ( x ) ≡ u ( x , 0 ) = a i sin( λ i x ) . i = 1 ◮ The coefficients a i in the eigenfunction expansion are found by multiplying both sides of the IC equation by sin( λ j x ) and integrating using the orthogonality property, i.e., PLOT � 1 2 λ i a i = f ( x ) sin( λ i x ) d x . λ i − sin( λ i ) cos( λ i ) 0

  71. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 ) Eigenvalues solution f ( λ ) f ( λ ) = tan( λ ) 3 2 1 1 3 5 7 9 2 π 2 π 2 π 2 π 2 π 0 λ π 2 π 3 π 4 π − 1 − 2 − 3 − 4

  72. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 ) Eigenvalues solution f ( λ ) f ( λ ) = tan( λ ) 3 2 1 1 3 5 7 9 2 π 2 π 2 π 2 π 2 π 0 λ π 2 π 3 π 4 π − 1 − 2 − 3 f ( λ ) = − λ h − 4

  73. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 ) Eigenvalues solution f ( λ ) f ( λ ) = tan( λ ) 3 2 1 1 3 5 7 9 2 π 2 π 2 π 2 π 2 π 0 λ π 2 π 3 π 4 π − 1 λ 1 − 2 λ 2 − 3 λ 3 f ( λ ) = − λ h − 4 λ 4 RETURN

  74. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 ) Initial shapes (i.e., t = 0 ) of four fundamental solutions X i ( x ) = sin( λ i x ) 1 X 3 ( x ) 0.75 X 1 ( x ) 0.5 0.25 0 x 0 0.25 0.5 0.75 1 -0.25 -0.5 -0.75 X 2 ( x ) X 4 ( x ) -1 RETURN

  75. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 , α = 1 , and f ( x ) = x 2 ) The shapes of four fundamental solutions scaled by the coefficients a i a i X i ( x ) = a i sin( λ i x ) 0.5 a 1 X 1 ( x ) 0.25 a 2 X 2 ( x ) a 3 X 3 ( x ) a 4 X 4 ( x ) 0 x 0 0.25 0.5 0.75 1 -0.25 -0.5

  76. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 , α = 1 , and f ( x ) = x 2 ) The final solution. (Notice that f ( x ) = x 2 does not satisfy the BC at x = 1 .) � 16 � 16 i α 2 t ) a i sin( λ i x ) exp( − λ 2 u ( x , t ) ≈ u i ( x , t ) = i = 1 i = 1 t = 0 (IC) : u ( x , 0 ) = x 2 1 0.75 0.5 0.25 0 x 0 0.25 0.5 0.75 1

  77. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 , α = 1 , and f ( x ) = x 2 ) The final solution. (Notice that f ( x ) = x 2 does not satisfy the BC at x = 1 .) � 16 � 16 i α 2 t ) a i sin( λ i x ) exp( − λ 2 u ( x , t ) ≈ u i ( x , t ) = i = 1 i = 1 t = 0 (IC) : u ( x , 0 ) = x 2 1 t = 0 . 000 0.75 0.5 0.25 0 x 0 0.25 0.5 0.75 1

  78. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 , α = 1 , and f ( x ) = x 2 ) The final solution. (Notice that f ( x ) = x 2 does not satisfy the BC at x = 1 .) � 16 � 16 i α 2 t ) a i sin( λ i x ) exp( − λ 2 u ( x , t ) ≈ u i ( x , t ) = i = 1 i = 1 t = 0 (IC) : u ( x , 0 ) = x 2 1 t = 0 . 001 0.75 0.5 0.25 0 x 0 0.25 0.5 0.75 1

  79. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 , α = 1 , and f ( x ) = x 2 ) The final solution. (Notice that f ( x ) = x 2 does not satisfy the BC at x = 1 .) � 16 � 16 i α 2 t ) a i sin( λ i x ) exp( − λ 2 u ( x , t ) ≈ u i ( x , t ) = i = 1 i = 1 t = 0 (IC) : u ( x , 0 ) = x 2 1 t = 0 . 002 0.75 0.5 0.25 0 x 0 0.25 0.5 0.75 1

  80. Introduction Classifications Canonical forms Separation of variables Example (results for h = 3 , α = 1 , and f ( x ) = x 2 ) The final solution. (Notice that f ( x ) = x 2 does not satisfy the BC at x = 1 .) � 16 � 16 i α 2 t ) a i sin( λ i x ) exp( − λ 2 u ( x , t ) ≈ u i ( x , t ) = i = 1 i = 1 t = 0 (IC) : u ( x , 0 ) = x 2 1 0.75 t = 0 . 005 0.5 0.25 0 x 0 0.25 0.5 0.75 1

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