Partial Differential Equation

1. Partial Differential Equation 3rd stage By Dr. Hero Waisi Salih Dr. Hero Waisi Salih

2. Chapter one Introduction Dr. Hero Waisi Salih

3. Definition /P.D.E. is an equation which involve two or more independent variables together with one or more dependent variables with their partial derivatives with respect to the variables. Definition/ the order of the P.D.E. is defined as order of the highest derivative occurring in equation Definition/ the power of the highest derivative appear in the P.D.E is called degree of P.D.E. Dr. Hero Waisi Salih

4. Monge’s symbol/ let Definition/ the P.D.E. is called linear if it is of the first degree of the dependent variables and the partial derivatives of the dependent variables occur in the eq. Definition/ the P.D.E. is called non- linear if it is not linear General form linear P.D.E. of the first order 2) linear P.D.E. of the second order Dr. Hero Waisi Salih

5. Definition/ P.D.E. is said to be quasi linear if it is linear in the heights order of derivative which appear in the equation. 1) is quasi linear P.D.E. of the first order. 2)is quasi linear P.D.E. of the second order. Definition/ an almost (half) linear P.D.E is quasi linear equation in which the coefficients of the highest order derivatives are functions only of the independent variables . 1) is almost linear P.D.E. of the first order. 2)is almost linear P.D.E. of the second order. Dr. Hero Waisi Salih

6. Origins of the P.D.E. 1) Elimination of arbitrary constant/ let is a function of independent variables and defined by relation Where are constants. We take differentiate of eq.(1) with respect to respectively, we obtain From (1),(2)&(3) to eliminate constants, we obtain P.D.E of the form Dr. Hero Waisi Salih

7. Example1/ eliminate arbitrary constant from Example2/ eliminate arbitrary constant from Example3/eliminate arbitrary constant from . 2) Elimination of arbitrary function/ suppose that are two independent functions of variables Let Where is an arbitrary function of We take differentiate eq(1) with respect to respectively we obtain Dr. Hero Waisi Salih

8. =0 =0 Or =0…….(2) =0…….(3) Dr. Hero Waisi Salih

9. To eliminate and we obtain Is the quasi linear of the first order or Dr. Hero Waisi Salih

10. Example4/eliminate the arbitrary ffrom the equation Example5/eliminate the arbitrary ffrom the equation Dr. Hero Waisi Salih

11. Origins of the second order Let …….(1) Is a general solution, where are arbitrary functions and are functions of we take differentiating of eq(1) with respect to respectively Dr. Hero Waisi Salih

12. Using Monge’s symbols Dr. Hero Waisi Salih

13. …(2) To eliminate from equation (2) We obtain the relation Dr. Hero Waisi Salih

14. We obtain the linear D.E of the second-order of the form Where are functions only of Dr. Hero Waisi Salih

15. Examples/ • find the P.D.E for is a g.s , where are arbitrary function • find the P.D.E for satisfied P.D.E is a g.s , where are arbitrary function • find the P.D.E for is a g.s , where are arbitrary function Dr. Hero Waisi Salih

16. Example/ find the P.D.E of the first – order of which complete integral is where are constants and is a parameter. Pfaffian Equation/ a D.E. of the form i.e, where are functions for all or some of independent variables for Dr. Hero Waisi Salih

17. Pfaffian Equation of three variables Where are functions of If we introduce the vector and Then eq(1) becomes Def/ a D.E. (1) is called exact if there exist a function such that , Now since then Dr. Hero Waisi Salih

18. And then , then Example/ Dr. Hero Waisi Salih

19. Def/ when multiply both sides of D.E By a suitable factor is an exact D.E, the equation is called integrable and is called integral factor of D.E i.e there exist function The function is called the primitive of D.E and is a general solution of D.E Dr. Hero Waisi Salih

20. Def/ if then Note/ if then the Pfaffian equation is exact Dr. Hero Waisi Salih