### A Novel Technique for solving non-linear Partial Differential Equation (PDE) under varying conditions

#### Abstract

*Artificial Neural Network (ANN), particularly radial basis function (RBF) is used to solve the Partial Differential Equations (PDE) instead of using explicit finite differences method (EFDM). Temperature distribution for an incompressible viscoelastic PTT fluid in a die is obtained on the basis of varying conditions of variables. Its result was compared with other methods. It is better than other methods (as less error, better in space and time complexity). *

#### References

Vidushi Sharma, Sachin Rai, Anurag Dev, “A Comprehensive Study of Artificial Neural Networks”, Volume 2, Issue 10, October 2012, ISSN: 2277 128X, International Journal of Advanced Research in Computer Science and Software Engineering.

S.I. Sulaiman, T.K. Abdul Rahman, and I. Musirin, “Optimizing One-hidden Layer Neural Network Design Using Evolutionary Programming”, 2009 5th International Colloquium on Signal Processing & Its Applications (CSPA).

Robert E. Uhrig, “INTRODUCTION TO ARTIFICIAL NEURAL NETWORKS”.

M.W. Gardner, S. R. Dorling, “Artificial Neural Network (The Multilayer Perceptron) The review of Applications in the atmospheric science”, Atmospheric environment vol. 32, No 14/15.

J. M. Ben´ıtez, J. L. Castro, and I. Requena, “Are Artificial Neural Networks Black Boxes?”. IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 8, NO. 5, SEPTEMBER 1997.

Ms. Sonali. B. Maind, “Research Paper on Basic of Artificial Neural Network”, International Journal on Recent and Innovation Trends in Computing and Communication ISSN: 2321-8169 Volume: 2 Issue: 1 96 – 100.

XIN YAO, SENIOR MEMBER, IEEE, “Evolving Artificial Neural Networks”, Invited Paper.

SHI Yu, ZHANG Jia-yin, “The Application of Artificial Neural Network Model in Estimation of Single Tree Volume Growth”.

W. Elliott Hutchcraft, Richard K. Gordon, “On the Effect of Order of Radial Basis Functions in the Solution of Partial Differential Equations”.

Li, Jianyu, Luo, Siwei, Qi, Yingjian, Huang, Yaping, “Numerical Solution of Elliptic Partial Differential Equation by Growing Radial Basis Function Neural Networks”.

I. Lagaris, A. Likas, and D. I. Fotiadis, “Artificial neural networks for solving ordinary and partial differential equations,” IEEE Trans. Neural Netw., vol. 9, no. 5, pp. 987–1000, Sep. 1998.

T. Cheng, F. L. Lewis, and M. Abu-Khalaf, “Fixed-final-time-constrained optimal control of nonlinear systems using neural network HJB approach,” IEEE Trans. Neural Netw., vol. 18, no. 6, pp. 1725–1737, Nov. 2007.

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford, U.K.: Clarendon Press, 1978.

T. J. R. Hughes, The Finite Element Method. Upper Saddle River, NJ, USA: Prentice-Hall, 1987.

W. Press, S. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing. Cambridge, U.K.: Cambridge Univ. Press, 1986.

C. Saloma, “Computational complexity and the observation of physical signals,” J. Appl. Phys., vol. 74, no. 9, pp. 5314-1–5314-6, 1993.

MATLAB Neural Network Toolbox, User’s Guide. Natick, MA, USA: MathWorks, 2005.

I. Lagaris, A. Likas, and D. Papageorgio, “Neural-network methods for boundary value problems with irregular boundaries,” IEEE Trans. Neural Netw., vol. 11, no. 5, pp. 1041–1049, Sep. 2000.

Keith Rudd, Gianluca Di Muro, and Silvia Ferrari, “A Constrained Backpropagation Approach for the Adaptive Solution of Partial Differential Equations”, IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 3, MARCH 2014.

E. J. Kansa, “Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics: I. Surface approximations and partial derivative estimates,” Comput. Math. Applicat., vol. 19, no. 6–8, pp. 127–145, 1990.

E. J. Kansa, “Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics: II. Solutions to parabolic, hyperbolic, and elliptic partial differential equations,” Comput. Math. Applicat., vol. 19, no. 6–8, pp. 147–161, 1990.

Nejib Smaoui, Suad Al-Enezi, “Modelling the dynamics of nonlinear partial di%erential equations using neural networks”, Journal of Computational and Applied Mathematics 170 (2004).

S., Reifb, K., & Unbehauenc, R. (2000). “Multilayer neural networks for solving a class of partial differential equations”. Neural Networks, 13, 385–396.

Jianyu, L., Siwei, L., Yingjian, Q., & Yaping, H. (2003). “Numerical solution of elliptic partial differential equation using radial basis function neural networks”. Neural Networks, 16, 729–734.

Shirvany, Y., Hayati, M., & Moradian, R. (2009). “Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations”. Applied Soft Computing, 9, 20–29.

Hornik, K., Stinchcombe, M., & White, H. (1989). “Multilayer feedforward networks are universal approximators”. Neural Networks, 2, 359–366.

Park, J., & Sandberg, I. W. (1991). “Universal approximation using radial basis function networks”. Neural Computing, 3(2), 246–257.

Schilling, R. J., Carroll, J. J., & Al-Ajlouni, A. F. (2001). “Approximation of nonlinear systems using radial basis function neural networks”. IEEE Transactions on Neural Networks, 12(1), 1–14.

Vesna Rankovic, Slobodan Savic, “Application of feedforward neural network in the study of dissociated gas flow along the porous wall”, Expert Systems with Applications 38 (2011) 12531–12536.

Shang, Z. (2005). “Application of artificial intelligence CFD based on neural network in vapor–water two-phase flow”. Engineering Applications of Artificial Intelligence, 18, 663–671.

Nam Mai-Duy, Thanh Tran-Cong, “Numaical solution of differential quations using multiquadric radial basis function networks” Neural NeMIorks 14,pp. 185-199,ZOOl.

Li Jianyu, Luo Siwei, Qi Yingjian and Huang Yaping, “Numerical solution of differential equations by radial basis function neural networks”.

Isaac Elias Lagaris, Aristidis Likas, Member, IEEE, and Dimitrios I. Fotiadis, “Artificial Neural Networks for Solving Ordinary and Partial Differential Equations”, IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 9, NO. 5, SEPTEMBER 1998.

Rehan Ali Shah, “MODELLING OF NON-NEWTONIAN FLUID PROBLEMS AND THEIR SOLUTIONS”, Ph.D. Thesis in Mathematics.

Omar S. Kasim, “Modifying Explicit Finite Difference Method by using Radial Basis Function Neural Network”, Ref J. of Comp & Math’s, vol. 10, No. 2, 2013.

پاراگلایدر Full Text: PDF

### Refbacks

- There are currently no refbacks.