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The Variational Iteration Method Which Should Be Followed
Nonlinear Science Letters A- Mathematics, Physics and Mechanics (2010)
  • Ji-Huan He, Donghua University
  • Guo-Cheng Wu, Donghua University
  • F. Austin, Hong Kong Polytechnic University

This paper proposes three standard variational iteration algorithms for solving differential equations, integro-differential equations, fractional differential equations, fractal differential equations, differential-difference equations and fractional/fractal differential-difference equations. The physical interpretations of the fractional calculus and the fractal derivative are given and an application to discrete lattice equations is discussed. The paper then examines the acceleration of some iteration formulae with particular emphasis being placed on the exponential Padé approximant that is suggested for solitary solutions and the sinusoidal Padé approximant that is usually used for periodic and compacton solutions. The paper points out that there may not be any physical meaning to the exact solutions of many nonlinear equations and stresses the importance of searching for approximate solutions that satisfy both the equations and the appropriate initial/boundary conditions. The variational iteration method is particularly suitable for solving this kind of problems. Approximate initial/boundary conditions and point boundary initial/conditions are also discussed, with the variational iteration method being capable of recovering the correct initial/boundary conditions and finding the solutions simultaneously. =============================================== Contents 1. Introduction 2. Variational Iteration Algorithm-I 3. Variational Iteration Algorithm-II 4. Variational Iteration Algorithm-III 5. Variational Iteration Algorithms for Ordinary Differential Equations and Partial Differential Equations 6. Variational Iteration Algorithms for Fractional Differential Equations 7. Physical Understanding of Fractional Calculus 8. Variational Iteration Algorithms for Fractal Differential Equations 9. Physical Understanding of Fractal Differential Equations 10. Variational Iteration Algorithms for Differential-difference Equations 11. Physical Understanding of Differential-difference Equations 12. Variational Iteration Algorithms for Fractal-difference Equations and Fractional-difference Equations 13. Series Solutions, Exponential Padé Approximant and Sinusoidal Padé Approximant 14. Approximate Solutions vs Exact Solutions 15. Approximate Initial/Boundary Conditions and Point Boundary Conditions 16. Conclusions

  • Variational iteration method; nonlinear equation; fractional differential equations; fractal differential equation; differential-difference equation; fractal differential-difference equation; fractal spacetime; porous flow; Lotka–Volterra equation; predator-prey model; solitary solutions; exponential Padé approximant; sinusoidal Padé approximant; approximate initial/boundary conditions; point boundary /initial conditions
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Ji-Huan He, Guo-Cheng Wu and F. Austin. "The Variational Iteration Method Which Should Be Followed" Nonlinear Science Letters A- Mathematics, Physics and Mechanics Vol. 1 Iss. 1 (2010)
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