Comparison set

Compare rates (English: comparison principle) are in the theory of differential equations important tools to be able to statements about the behavior of solutions to meet these equations. These are particularly important because it is often not possible to give explicit solution formulas for such equations.

Comparative theorem for ordinary differential equations

In the theory of ordinary differential equations , the comparison theorem is one of the most important tools for making statements about solutions of (scalar) differential equations of the first order that cannot be calculated explicitly.

It clearly means that solutions of the same differential equation remain arranged, i. i.e., is for two solutions of a scalar differential equation, then remains on the entire common domain. In particular, if a solution of the differential equation is explicitly known, then estimates for solutions that cannot be explicitly calculated can be obtained from it. ${\ displaystyle u (a) <v (a)}$ ${\ displaystyle u (x) <v (x)}$

However, since it is not always possible to find explicit solutions, it is necessary for practical reasons to be able to compare with upper and lower solutions, as these are easier to construct.

formulation

Let it be , continuous and locally Lipschitz continuous in the second variable. Next were an upper and lower solution , that is, it applies to all of ${\ displaystyle D \ subset \ mathbb {R}}$ ${\ displaystyle F: (a, b] \ times D \ rightarrow \ mathbb {R}}$ ${\ displaystyle y _ {+}, y _ {-} \ in C ([a, b]) \ cap C ^ {1} ((a, b])}$ ${\ displaystyle \ y '= F (x, y)}$ ${\ displaystyle y _ {+} (x), y _ {-} (x) \ in D}$ ${\ displaystyle x \ in (a, b]}$

{\ displaystyle y _ {+} '(x) \ geq F (x, y _ {+} (x)) \ {\ textrm {and}} \ y _ {-}' (x) \ leq F (x, y_ { -} (x))}

for everyone . In addition , it follows ${\ displaystyle x \ in (a, b]}$ ${\ displaystyle \ y _ {+} (a)> y _ {-} (a)}$

{\ displaystyle \ y _ {+} (x)> y _ {-} (x)}

for everyone . ${\ displaystyle x \ in [a, b]}$

variant

The same applies to yield by replacing: If it follows for all . ${\ displaystyle (a, b]}$ ${\ displaystyle [a, b)}$ ${\ displaystyle \ y _ {+} (b) <y _ {-} (b)}$ ${\ displaystyle \ y _ {+} (x) <y _ {-} (x)}$ ${\ displaystyle x \ in [a, b]}$

proof

Be and . Suppose . For follows on and . Fix one . It's a compact subset of . Since locally Lipschitz continuous in the second variable, there is one with ${\ displaystyle \ d (x): = y _ {+} (x) -y _ {-} (x)}$ ${\ displaystyle A: = \ {x \ in [a, b] \ | \ d (x) \ leq 0 \}}$ ${\ displaystyle A \ neq \ emptyset}$ ${\ displaystyle x_ {0}: = \ min A> a}$ ${\ displaystyle d> 0}$ ${\ displaystyle [a, x_ {0})}$ ${\ displaystyle d (x_ {0}) = 0}$ ${\ displaystyle s_ {0} \ in (a, x_ {0})}$ ${\ displaystyle K: = \ {y _ {+} (x) \ | \ x \ in [s_ {0}, x_ {0}] \} \ cup \ {y _ {-} (x) \ | \ x \ in [s_ {0}, x_ {0}] \}}$ ${\ displaystyle D}$ ${\ displaystyle F}$ ${\ displaystyle L \ geq 0}$

{\ displaystyle \ | F (x, y) -F (x, z) \ | \ leq L \ | yz \ |}

for everyone and . It follows ${\ displaystyle x \ in [s_ {0}, x_ {0}]}$ ${\ displaystyle y, z \ in K}$

{\ displaystyle d '(x) = y _ {+}' (x) -y _ {-} '(x) \ geq F (x, y _ {+} (x)) - F (x, y _ {-} ( x)) \ geq -L \ | y _ {+} (x) -y _ {-} (x) \ | = -L \ cdot d (x)}

for everyone , therefore for everyone . Integration delivers , i.e. for everyone . The contradiction follows from the continuity of . ${\ displaystyle x \ in [s_ {0}, x_ {0}]}$ ${\ displaystyle {\ frac {d '(x)} {d (x)}} \ geq -L}$ ${\ displaystyle x \ in [s_ {0}, x_ {0})}$ ${\ displaystyle \ ln d (x) - \ ln d (s_ {0}) \ geq -L (x-s_ {0})}$ ${\ displaystyle d (x) \ geq d (s_ {0}) e ^ {- L (x-s_ {0})}}$ ${\ displaystyle x \ in (s_ {0}, x_ {0})}$ ${\ displaystyle d}$ ${\ displaystyle d (x_ {0}) \ geq d (s_ {0}) e ^ {- L (x_ {0} -s_ {0})}> 0}$

{\ displaystyle \ Box}

example

Consider the initial value problem

{\ displaystyle y '= {\ frac {y ^ {6} -64} {1 + x ^ {2} y ^ {2}}} \, \ y (0) = 1 \.}

It has a non-continuable solution . The differential equation has the trivial solutions and . According to the comparative principle, applied to and , applies to all . In particular, it follows from the theorem about the maximum existence interval that , that is, the solution exists globally. In addition, the estimate provides . Hence it is strictly monotonically decreasing. ${\ displaystyle y: D \ rightarrow \ mathbb {R}}$ ${\ displaystyle y_ {1} (x): \ equiv 2}$ ${\ displaystyle y_ {2} (x): \ equiv -2}$ ${\ displaystyle \ y, y_ {1}}$ ${\ displaystyle \ y, y_ {2}}$ ${\ displaystyle \ -2 <y (x) <2}$ ${\ displaystyle x \ in D}$ ${\ displaystyle D = \ mathbb {R}}$ ${\ displaystyle y '<0}$ ${\ displaystyle y}$

{\ displaystyle \ Box}

Comparative theorems for partial differential equations

Comparative theorems also exist for partial differential equations , e.g. for the nonlinear parabolic differential equation . As a generalization of the weak maximum principle , the comparative theorems allow statements to be made in particular about the solutions to nonlinear partial differential equations.

literature

Wolfgang Walter: Ordinary differential equations . 6th edition. Springer-Verlag Berlin / Heidelberg / New York 1996, ISBN 3-540-59038-2 .

Individual evidence

↑ Gerhard Dziuk: Theory and numerics of partial differential equations. de Gruyter, Berlin 2010, ISBN 978-3-11-014843-5 , pages 190-194

[1] Gerhard Dziuk: Theory and numerics of partial differential equations. de Gruyter, Berlin 2010, ISBN 978-3-11-014843-5 , pages 190-194