This document discusses the classification of almost linear second order partial differential equations with n independent variables. It defines an almost linear second order PDE and introduces the characteristic form and principal part of the operator. The classification is based on the canonical form of the characteristic form and includes elliptic, hyperbolic, ultra hyperbolic, and parabolic types. Elliptic operators have a definite characteristic form, while hyperbolic and ultra hyperbolic have indefinite forms. Parabolic operators have a singular characteristic form matrix. Examples of the normal forms for different types are provided.

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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