Modified Variational Iteration Method for Partial Differential Equations Using Ma’s Transformation

1. Modified Variational Iteration Method for Partial Differential Equations Using Ma’s Transformation SYED TAUSEEF MOHYUD-DIN

2. Variational Iteration Techniques for Solving Initial and Boundary Value Problems Introduction and History Use of Initial and Boundary Conditions Correction Functional Restricted Variation Selection of Initial Value Conversion to a System of Equations Identification of Lagrange Multiplier Simpler

3. Variational Iteration Techniques for Solving Initial and Boundary Value Problems • Applications of Variational Iteration Method • Modifications (VIMHP and VIMAP) • Applications in Singular Problems (Use of New Transformations)

4. Advantages of Variational Iteration Method • Use of Lagrange Multiplier (reduces the successive applications of integral operator) • Independent of the Complexities of Adomian’s Polynomials • Use of Initial Conditions only • No Discretization or Linearization or Unrealistic Assumptions • Independent of the Small Parameter Assumption

5. Applications • Boundary Value Problems of various-orders • Boussinesq Equations • Thomas-Fermi Model • Unsteady Flow of Gas through Porous Medium • Boundary Layer Flows • Blasius Problem • Goursat Problems • Laplace Problems

6. Applications • Heat and Wave Like Models • Burger Equations • Parabolic Equations • KdVs of Third, Fourth and Seventh-orders • Evolution Equations • Higher-dimensional IBVPS • Helmholtz Equations

7. Applications • Fisher’s Equations • Schrödinger Equations • Sine-Gordon Equations • Telegraph Equations • Flierl Petviashivili Equations • Lane-Emden Equations • Emden-Fowler Equations

8. Variational Iteration Method Correction functional

9. Variational Iteration Method Using He’s Polynomials (VIMHP)

10. Modified Variational Iteration Method for Partial Differential Equations Using Ma’s Transformation

11. Helmholtz Equation with initial conditions The exact solution

12. ) (by setting Applying Ma’s transformation with The correction functional

13. Applying modified variational iteration method (MVIM)

14. Comparing the co-efficient of like powers of p, following approximants are obtained .

15. The series solution The inverse transformation

16. the use of initial condition The solution after two iterations is given by

17. Figure 3.1 Solution by Proposed Algorithm Exact solution

18. Helmholtz Equation with initial conditions The exact solution for this problem is

19. Applying Ma’s transformation (by setting with The correction functional is given by

20. Applying modified variational iteration method (MVIM)

21. Comparing the co-efficient of like powers of p, following approximants are obtained .

22. The series solution is given by the inverse transformation will yield

23. The use of initial condition gives The solution after two iterations is given by

24. Table 1 Table 1 (Error estimates at ) *Error= Exact solution – Approximate solution

25. Homogeneous Telegraph Equation. with initial and boundary conditions The exact solution for this problem is

26. (by setting Applying Ma’s transformation with

27. Applying modified variational iteration method (MVIM)

28. Comparing the co-efficient of like powers of p, following approximants are obtained The series solution is given by

29. The inverse transformation would yield and use of initial condition gives

30. The solution after two iterations is given by . Solution by Proposed Algorithm Exact solution

31. CONCLUSION

32. THANK YOU