THE Q-HOMOTOPY ANALYSIS METHOD (Q-HAM) - IJAMM

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60 Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012. M. A. El-Tawil and S. N. Huseen Figure : The Absolute error of of q-HAM ( for problem (10) at and using Figure : The Absolute error of of q-HAM ( for problem (10) at and using 3.2 The Riccati equation Let Absolute Error 0.0012 0.0010 0.0008 0.0006 0.0004 0.0002 Absolute Error 30 000 25 000 20 000 15 000 10 000 5000 The exact solution is known to be 1 2 3 4 8 10 12 t t A.E n 1 A.E n 40 A.E n 1 A.E n 40 (18) (19)
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012. The q-Homotopy Analysis Method For q- HAM solution we choose the linear operator : [ ] With the property : [ ] Where is constant. Using initial approximation We define a nonlinear operator as : [ ] We construct the zero order deformation equation [ ] [ ] Where . We can take H , and the order deformation equation is : [ ] (21) With the initial conditions for : Where: and { ∑ [ ] Now the solution of equation (18) for becomes: ∫ Where the constant of integration is determined by the initial conditions (22). We now obtain components of the solution using q- HAM as follows: , can be calculated similarly.Then the series solution expression by q- HAM can be written in the form: (20) (22) 61
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THE Q-HOMOTOPY ANALYSIS METHOD (Q-HAM) - IJAMM

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