On Combining with Fourier Transform and Adomian Methods to solve the Riccati Equations

Öz

Bu makalede, Riccati denklemlerini çözmek için Fourier dönüşüm yöntemini ile Adomian ayrıştırma yöntemi uyguluyoruz. Önerilen yöntem, Fourier dönüşümü ve adomian ayrıştırma yöntemlerine dayanmaktadır. FADM'ler kullanılarak elde edilen çözümler, Runge Kutta2 ve Euler yöntemi kullanılarak elde edilen sayısal çözümlerle karşılaştırılmıştır. Ayrıca çözümlerin hata grafikleri sunulmuştur.

Anahtar Kelimeler

• Fourier transform method,
• Adomian decomposition method
• numerical solution of ODE

On Combining with Fourier Transform and Adomian Methods to solve the Riccati Equations

Öz

In this paper, we apply the Fourier transform method with the Adomian decomposition method to solve Riccati equations . Proposed method is based on the Fourier transform and adomian decomposition methods. The solutions obtained using FADMs are compared with the numerical solutions obtained using the Rung Kutta2 and Euler method. Also, error graphs of the solutions are presented.

Anahtar Kelimeler

• Fourier transform method
• Adomian decomposition method
• numerical solution of ODE

Kaynakça

1. [1] Changqing Y, Jianhua H, Beibo Q (2012) Numerical solution of Riccati differential equations by using hybrid functions and tau method. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 6, 871-874
2. [2] Gemechis F, Tesfaye A (2016) Numerical solution of quadratic Riccati differential equations, Egyptian journal of basic and applied sciences 3 392–397
3. [3] Fateme G, Esmaile K (2017) Approximate solution for quadratic Riccati differential equation, Journal of Taibah University for Science 11 246–250
4. [4] Jafar B, Mohsen D (2015) Numerical Solution of Riccati Equations by the Adomian and Asymptotic Decomposition Methods over Extended Domains. International Journal of Differential Equations, Article ID 580741, 7 pages
5. [5] Magdy A, Ahmed A, Ahmed A (2004) Solving Riccati differential equation using Adomians decomposition method. Applied Mathematics and Computation 157 503–514
6. [6] B. Gbadamosi , O. Adebimpe , E.I. Akinola , I.A. Olopade (2012) Solving riccati equation using adomian decomposition method, International Journal of Pure and Applied Mathematics 78 , 409-417
7. [7] B. Batiha, M. S. M. Noorani and I. Hashim( 2007) “ Application of Variational Iteration Method to a General Riccati Equation, International Mathematical Forum 2 2759 – 2770
8. [8] Supriya Mukherjee1 ∗ , Banamali Roy, Solution of Riccati Equation with Variable Co-efficient by Differential Transform Method, International Journal of Nonlinear Science Vol.14(2012) No.2,pp.251-256

9. [9] Bracewell, R. N.( 2000) The Fourier Transform and Its Applications (3rd bas.), Boston: McGraw-Hill Book Campany,
10. [10] Francis J. N, Albert B. (2001), Afirst Course in Wavelets with Fourier Analysis, Wiley Edition.
11. [11] Murray R.S(1974) Fourier Analysis, Schaum.s Outline Series, McGraw Hill,
12. [12] Tarig E(2012) Solution of Nonlinear Differential Equations Using Mixture of Elzaki Transform and Differential Transform Method, International Mathematical Forum, 7, 631-638 , [13] Tarig. M, Eman H ( 2012) Homotopy Perturbation and Elzaki Transform for Solving Nonlinear Partial Differential Equations, Mathematical Theory and Modeling 2
13. [14] Yusufoğlu, E., .Numerical Solution of duffing equation by the Laplace decomposition algorithm., Applied Mathematics and Computation,177 (2): 572-580 (2006).
14. [15] Chapra, Steven C., Numerical methods for engineers / Steven C. Chapra, Raymond P. Canale.6th ed., McGraw-Hill, (2010)
15. [16] Steven T. Karris, Numerical Analysis Using MATLAB and Excell, Therd Edition Orchard Publications, Fremont, California, (2007)