DELIB

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DELIB

After specifying a differential equation (there are several different ways to specify one, see below) the differential equation library will return information and/or references for that differential equation.

1. Ordinary differential equations    f(x,y,y',y'',...)=0
   1. First order equations    f(x,y,y')=0
      • Linear equations    a(x)y'+b(x)y+c(x)=d(x)
      • Non-linear equations    NOT a(x)y'+b(x)y+c(x)=d(x)
   2. Second order equations    f(x,y,y',y'')=0
      • Linear equations    a(x)y''+b(x)y'+c(x)y+d(x)=e(x)
      • Non-linear equations    NOT a(x)y''+b(x)y'+c(x)y+d(x)=e(x)
   3. Higher order equations ==> (Linear equations    Non-linear equations)
      
   4. Variable order equations
      
   5. Systems of ordinary differential equations ==> (All linear equations    Some non-linear equations)
2. Partial differential equations
   1. PDEs of the form:  A(x,t)u_t+B(x,t)u_x=0   A(x,t)u_t+B(x,t)u_xx=0   A(x,t)u_tt+B(x,t)u_xx=0   ∇²u=0   ∇²u- Λu=0   ...
   2. Linear equations ==> (Two variables    More than two variables)
      
   3. Non-linear equations ==> ( Second order non-linearity    Higher order non-linearity    Variable order non-linearity)
   4. Systems of partial differential equations ==> (All linear equations    Some non-linear equations)
      
3. Equations defined by Hamiltonians ==> (Linear equations    Non-linear equations)
   
4. Other types of equations ==> (Differential algebraic equations    Differential-delay equations    Fractional differential equations    Stochastic differential equations)

EXAMPLE The Airy equation is found in: "Ordinary differential equations", "Second order equations", "Linear equations" section.

EXAMPLE For the Airy equation, click on "A".

• BASIC FEATURES
  • Type:    ODE  PDE  unspecified
  • Size:    Single equation  Many equations  unspecified
• ADVANCED FEATURES
  • Mathematics involved:    vectors  matrices  tensors  Christoffel symbols
  • Operators involved:    Del operator  Laplacian  D'Alembert
  • Operations involved:    Dot product  Cross product
  • Only match DES that have (and DELIB knows):     Backlund transformation  Hamiltonian  Lax pair  exact solution  known Lie group  ICs and/or BCs

• Following features are for:    Entire DES  A single DE in the DES  A single BC or IC  A single term
• Is linear:     Yes  No  unspecified
• Number of terms:     1 term  2 terms  3 terms  4 terms  unspecified
• Highest order:     First order  Second order  Third order  unspecified order
• Highest degree:     1  2  3  n  unspecified degree
• Number of independent variables:     1  2  3  n  unspecified
• Number of dependent variables:     1  2  3  n  unspecified
• Contains:     pure derivative  mixed derivative  un-expanded operator  arbitrary function  powers of dependent variables  powers of independent variables
• Appearance of independent variable:     constants  polynomial expressions  unspecified
• Constants that appear: e i pi square root of 2 square root of pi ...
• Functions that appear: powers square root sin cos tan csc sec cot sinh cosh tanh csch sech coth ...