Modified Variational Iteration Method for Partial Differential Equations

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

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

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

Advantages of Variational Iteration Method o o o 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

Applications o o o o 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

Applications o o o o Heat and Wave Like Models Burger Equations Parabolic Equations Kd. Vs of Third, Fourth and Seventh-orders Evolution Equations Higher-dimensional IBVPS Helmholtz Equations

Applications o o o o Fisher’s Equations Schrödinger Equations Sine-Gordon Equations Telegraph Equations Flierl Petviashivili Equations Lane-Emden Equations Emden-Fowler Equations

Variational Iteration Method Correction functional

Variational Iteration Method Using He’s Polynomials (VIMHP)

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

Helmholtz Equation with initial conditions The exact solution

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

Applying modified variational iteration method (MVIM)

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

The series solution The inverse transformation

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

Figure 3. 1 Solution by Proposed Algorithm Exact solution

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

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

Applying modified variational iteration method (MVIM)

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

The series solution is given by the inverse transformation will yield

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

Table 1 (Error estimates at *Error = ) Exact solution Approx solution -1. 0 -. 0744491770 -. 082675613 8. 22 E-03 -0. 8 -. 0039143995 -. 0058010496 1. 88 E-03 -0. 6 . 0722477834 . 0719893726 2. 58 E-04 -0. 4 . 1384269365 . 1384142557 1. 26 E-05 -0. 2 . 1829867759 . 1829865713 2. 04 E-07 0 . 1986693308 0. 000000 0. 2 . 1829991064 . 1829865713 1. 25 E-05 0. 6 . 1386872460 . 1384142557 2. 72 E-04 0. 8 . 0740356935 . 0719893726 2. 04 E-03 1. 0 . 0033413560 -. 0058010496 9. 14 E-03 1. 0 -. 0526997339 -. 0826756135 2. 99 E-02 Exact solution – Approximate solution *Errors

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

Applying Ma’s transformation with (by setting

Applying modified variational iteration method (MVIM)

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

The inverse transformation would yield and use of initial condition gives

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

CONCLUSION

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