Classification of Almost Linear Equation in n-Independent Variables

2. Classification of Almost
Linear Equation in
n-Independent Variables
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4. Layout
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5. INTRODUCTION:
Let 𝑢 = 𝑢(𝑥1, . . , 𝑥𝑛) be a function of n-independent variable 𝑥1, … , 𝑥𝑛.
A partial differential equation that contain the independent variable
𝑥1, … , 𝑥𝑛 , the dependent variable or the unknown function u and its
partial derivatives up to some order . It has the form
𝐹 𝑥1, … , 𝑥𝑛, 𝑢, 𝑢𝑥1
, . . , 𝑢𝑥𝑛
, 𝑢𝑥1𝑥1
, … . , 𝑢𝑥𝑖𝑥𝑗
, … = 0
Where F is given function and 𝑢𝑥𝑗
=
𝜕𝑢
𝜕𝑥𝑗
, 𝑢𝑥𝑖𝑥𝑗
=
𝜕2𝑢
𝜕𝑥𝑖𝜕𝑥𝑗
𝑖, 𝑗 = 1, … , 𝑛
are the partial derivatives of u. The order of partial differential equation
is the order of the highest derivatives which is appear in the equation.
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6. Almost Linear Second Order
Equation In n-Independent Variables:
An almost-linear second order equation in n-independent variables 𝑥1, 𝑥2, … . . 𝑥𝑛 is
of the form
𝑖=1
𝑛
𝑗=1
𝑛
𝐴𝑖𝑗
𝜕2𝑢
𝜕𝑥𝑖𝜕𝑥𝑗
+ M 𝑥1, 𝑥2, … . 𝑥𝑛, 𝑢, 𝑢𝑥1
, 𝑢𝑥2
, … , 𝑢𝑥𝑛
= 0 (1)
It is assumed that the coefficient 𝐴𝑖𝑗 are real-valued continuously differentiable
function of 𝑥1, 𝑥2, … . , 𝑥𝑛 and that 𝐴𝑖𝑗 = 𝐴𝑗𝑖, i,j=1,2,…,n.
The linear operator
𝐿 = 𝑖=1
𝑛
𝑖=1
𝑛
𝐴𝑖𝑗𝐷𝑥𝑖
𝐷𝑥𝑗
where 𝐷𝑥𝑖
=
𝜕
𝜕𝑥𝑖
𝑖 = 1,2, … . , 𝑛
𝐷𝑥𝑗
=
𝜕
𝜕𝑥𝑗
𝑗 = 1,2, … . , 𝑛
is called the principal part of the operator L appearing in equation (1).
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7. Classification based on the
characteristic form:
Classification based on the characteristic form
𝑄 𝜉 = 𝑖=1
𝑛
𝑖=1
𝑛
𝐴𝑖𝑗𝜉𝑖𝜉𝑗 (2)
It is understood that the function 𝐴𝑖𝑗 are evaluated at 𝑥1 = 𝑥10, … … . , 𝑥𝑛 = 𝑥𝑛0 and
(𝜉1, … . , 𝜉𝑛) is a real n-tuple.
A well known property of such a real quadratic is that there exist a linear
transformation 𝜉𝑖 = 𝑗=1
𝑛
𝑆𝑖𝑗𝜂𝑗 i=1,2,…,n where 𝑆 = 𝑆𝑖𝑗 is a non-singular matrix such
that 𝑄(𝜉) is reduce to canonical form.
𝑄 𝜂 = 𝜂1
2
+ ⋯ + 𝜂𝑝
2
− 𝜂𝑝+1
2
− ⋯ − 𝜂𝑝+𝑞
2
(3)
Where 𝜂𝑖 ≠ 0, 𝑖 = 1,2, … , 𝑝 + 𝑞
𝑝 ≥ 0 is called the positive index in (3).
𝑞 ≤ 0 is called the negative index.
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8. Continue…..
Therefore the classification of the canonical form of the characteristic form of
equation (2).
Let
𝜉 = 𝐴𝜂
be the non-singular linear transformation reducing the equation (2) to the
canonical form. Then the transformation
𝑦 = 𝐴𝑇
𝑥
reducing equation (3) ⇒
𝑖=1
𝑝
𝑣𝑦𝑖𝑦𝑖
− 𝑖=1
𝑞
𝑣𝑦𝑝+𝑖𝑦𝑝+𝑖 + 𝐹 𝑦, 𝑣, 𝛻𝑣 = 0,
Where
𝑣 𝑦 = 𝑢( 𝐴𝑇 −1𝑦).
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9. Continue…..
And the number 𝑟 = 𝑝 + 𝑞 is called rank of characteristic form Q at a point
𝑥1 = 𝑥10, … 𝑥𝑛 = 𝑥𝑛0. The rank 𝑟 ≤ 𝑛 , and r is called the rank of matrix 𝐴 = 𝐴𝑖𝑗 at the
point. The number v = 𝑛 − 𝑟 is called the nullity of the characteristic form, and 𝜈 ≥ 0.
Thus 𝜈>0 if and only if the rank of matrix A is less then n that is if and only if A is a
singular matrix.
The important thing is that these numbers are invariant with respect to real
nonsingular linear transformations of the variable 𝜉1, 𝜉2 , … , 𝜉𝑛 that is they have the
same value regardless the mode of the reduction of the 𝑄 𝜉 to the form in eqn (3).
At 𝑥1 = 𝑥10, … , 𝑥𝑛 = 𝑥𝑛0 the operator L (in equ 1) is said to be
1. Elliptic if 𝜈 = 0, 𝑎𝑛𝑑 𝑒𝑖𝑡ℎ𝑒𝑟 𝑝 = 0 𝑜𝑟 𝑞 = 0
2. Hyperbolic if 𝜈 = 0 𝑎𝑛𝑑 𝑒𝑖𝑡ℎ𝑒𝑟 𝑝 = 𝑛 − 1 𝑎𝑛𝑑 𝑞 = 1 𝑜𝑟 𝑝 = 1 𝑎𝑛𝑑 𝑞 = 𝑛 − 1
3. Ultra hyperbolic if 𝜈 = 0 𝑎𝑛𝑑 1 < 𝑞 < 𝑛 − 1 (𝑠𝑜 1 < 𝑝 < 𝑛 − 1 )
4. Parabolic if 𝜈 > 0
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10. Continue…
• The operator L is elliptic at the point if and only if ,the characteristic form is definite ,
being either positive definite or negative definite .
• The form is positive definite if 𝑄(𝜉) ≥ 0 holds for all real n-tuples ( 𝜉1, . . , 𝜉𝑛 )
and 𝑄 𝜉 = 0 if and only if 𝜉1 = ⋯ = 𝜉𝑛 = 0.
• The form is the negative definite if
𝑄(𝜉) ≤ 0 .
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11. Explanation.
Question:
𝑄 𝜉 = 𝑖=1
𝑛
𝑗=1
𝑛
𝐴𝑖𝑗𝜉𝑖𝜉𝑗 (2)
We discuss this general form for n=2 and for n > 2.
Solution:
Case:1
For n=2
𝑄 𝜉 = 𝐴11𝜉1
2
+ 2𝐴12𝜉1𝜉2 + 𝐴22𝜉2
2
This form is definite if and only if Δ = 𝐴12
2
− 𝐴11𝐴22 < 0.
Thus the criterion for ellipticity of L in the general case reduce in the particular case
n=2 to the criterion stated previously for the operator.
In the same way it follows that the criteria for the hyperbolicity and parabolicity
state above reduce to Δ > 0 and Δ = 0 respectively.
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12. continue….
• Case :2
If n>2, it is not possible in general to reduce Equ (1) to normal form in a
region. However in the special case where the coefficient 𝐴𝑖𝑗 are constant it is possible
to reduce the differential equation to normal form. Then equation (1) becomes
1. If L is Elliptic
Δ𝑣 + 𝑐𝑣 = 𝐹(𝑥1, … , 𝑥𝑛)
where ‘c’ is constant and Δ𝑣 =
𝜕2𝑣
𝜕𝑥1
2 + ⋯ +
𝜕2𝑣
𝜕𝑥𝑛
2
2. If L is hyperbolic
𝜕2𝑣
𝜕𝑥1
2 + ⋯ +
𝜕2𝑣
𝜕𝑥𝑛−1
2 −
𝜕2𝑣
𝜕𝑥𝑛
2 + 𝑐𝑣 = 𝐹(𝑥1, … , 𝑥𝑛)
3. If L is ultra hyperbolic the equation ca be reduce to normal form
𝜕2𝑣
𝜕𝑥1
2 + ⋯ +
𝜕2𝑣
𝜕𝑥𝑝
2 −
𝜕2𝑣
𝜕𝑥𝑝+1
2 − ⋯ −
𝜕2𝑣
𝜕𝑥𝑝+𝑞
2 + 𝑐𝑣 = 𝐹(𝑥1, … , 𝑥𝑛)
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13. continue….
Where 1 < 𝑝 < 𝑛 − 1 𝑎𝑛𝑑 𝑝 + 𝑞 = 𝑛 .
4. If L is parabolic
𝜕2𝑣
𝜕𝑥1
2 + ⋯ +
𝜕2𝑣
𝜕𝑥𝑟
2 + 𝐵𝑟+1
𝜕𝑣
𝜕𝑥𝑟+1
2 + ⋯ + 𝐵𝑛
𝜕𝑣
𝜕𝑥𝑛
+ 𝑐𝑣 = 𝐹(𝑥1, . . . , 𝑥𝑛)
where 0 < 𝑟 < 𝑛.
If n=2 the normal written above in the hyperbolic case is
𝜕2𝑣
𝜕𝑥2 −
𝜕2𝑣
𝜕𝑦2 + 𝑐𝑣 = 𝐹 𝑥, 𝑦 .
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14. Example:
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15. Continue….
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16. Continue…
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19. Explanation…..
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