In this paper, a numerical solution to the Troesch problem using a memetic computing technique is proposed. In this regard, a linear combination of Gaussian radial basis functions (GRBF) as approximate mathematical framework is suggested. A set of unknown yet adaptable parameters are modeled in a cost function, representing the trial solution. Differential Evolution (DE) algorithm crossed with Interior Point algorithm (IPA) and Pattern Search (PS) are utilized to find the unknowns. Numerical results based on three special cases of proposed scheme are compared with the exact solution as well as many other numerical and classical approximation methods. Analysis reveals that the proposed scheme is promising in terms of accuracy and conformation to the exact solution.
S.H.Mirmoradi, I.Hosseinpour and S.Ghanbarpour, A.Barari, “Application of an approximate analytical method to nonlinear Troesch’s problem,”Applied Mathematical Sciences3(32) (2009), 1579–1585.
2.
H.N.Hassan and M.A.El-Tawil, “An efficient analytic approach for solving two-point nonlinear boundary value problems by homotopy analysis method,”Mathematical Methods in The Applied Sciences34 (2011), 977–989.
3.
S.Momani, S.Abuasad and Z.Odibat, “Variational iteration method for solving nonlinear boundary value problems,”Applied Mathematics and Computation 183 (2006), 1351-1358.
4.
E.Deeba and S.A.Khuri, S.Xie, ”An Algorithm for Solving Boundary Value Problems,”Journal of Computational Physics159 (2000), 125–138.
5.
S.H.Chang, “Numerical solution of Troesch’s problem by simple shooting method,”Applied Mathematics and Computation216 (2010), 3303–3306.
6.
S.T.Mohyu-din, “Solution of Troesch’s problem using He’s polynomials,”Revista De La, Union Mathematica Argentina52(1) (2011), 143–148.
7.
X.Feng, L.Mei and G.He, “An efficient algorithm for solving Troesch’s problem,”Applied Mathematics and Computation189 (2007), 500–507.
8.
F.Geng and M.Cui, “A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM,”Applied Mathematics and Computation217 (2011), 4676–4681.
9.
S.A.Khuri and A.Sayfy, “Troesch’s problem: A B-spline collocation approach,”Mathematical and Computer Modeling54 (2011), 1907-1918.
10.
E.S.Weibel and R.K.M.Landshoff, “The Plasma in magnetic field,”Stanford university press, Stanford, 130 (1958).
11.
B.A.Troesch, ”A simple approach to a sensitive two-point boundary value problem,”Journal of Computational Physics21 (1976), 279—290.
12.
J.A.Khan, R.M.A.Zahoor, and I.M.Qureshi, “Swarm intelligence for the problems of non-linear ordinary differential equations and its application to well-known Wessinger’s equation,”European Journal of Scientific Research34(4) (2009), 514-525.
13.
M.A.Behrang, M.Ghalambaz, E.Assareh and A.R.Noghrehabadi, “A new solution for natural convection of darcian fluid about a vertical full cone embedded in porous media prescribed wall temperature by using a hybrid neural network-particle swarm optimization method,”World Academy of Science, Engineering and Technology49 (2011), 1098-1103.
14.
S.A.Malik, I.M.Qureshi, M.Zubair and I.Haq, “Solution to force-free and forced duffing-van der pol oscillator using memetic computing,”Journal of Basic and Applied Scientific Research2(11) (2012), 11136-11148.
15.
R.M.A.Zahoor, J.A.Khan and I.M.Qureshi, “Evolutionary computation technique for solving Riccati differential equation of arbitrary order,”World Academy of Science, Engineering and Technology48 (2009), 303-309.
16.
R.Storn, K.Price, “DE-a simple and efficient heuristic for global optimization over continuous space”, Journal of Global Optimization114(4) (1997), 341-359.
17.
G.Lesaja, “Introducing interior-point methods for introductory operations research courses and/or linear programming courses,”The Open Operational Research Journal3 (2009), 1-12.
18.
A.K.Al-Othman and K.M.EL-Nagger, “Application of pattern search method to power system security constrained economic dispatch,”International Journal of Engineering and Applied Sciences B (1) (2008), 25-30.
19.
V.Torczon, “On the convergence of pattern search algorithms,”SIAM Journal of Optimization7(1) (1997), 1-25.
20.
Garg,P, “A comparison between memetic algorithm and genetic algorithm for the cryptanalysis of simplified data encryption standard algorithm.”International Journal of Network Security & Its Applications (IJNSA)1(1) (2009), 34-42.
21.
S.A.Malik, I.M.Qureshi, M.Zubair, M.Amir, “Numerical Solution to Troesch’s Problem Using Hybrid Heuristic Computing”, J Basic Appl Sci Res3(7) (2013), 10-16.
22.
Atta-ur-Rahman, M.Mahmud, K.Sultan, N.Aldhafferi, A.Alqahtani, D.Abdullah, “Medical Image Watermarking for Fragility and Robustness: A Chaos, ECC and RRNS Based Approach”, Journal of Medical Imaging and Health Informatics8(6) (2018), 1192-1200.
23.
Storn,R. and Price,K. (1997) Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces, Journal of Global Optimization11, pp. 341–359.
24.
Atta-ur-Rahman, QureshiI.M., MalikA.N. and NaseemM.T., “A Real Time Adaptive Resource Allocation Scheme for OFDM Systems using GRBF-Neural Networks and Fuzzy Rule Base System”, International Arab Journal of Information Technology (IAJIT)11(6) (2014), 593-601.
25.
Atta-ur-Rahman, QureshiI.M., MalikA.N. and NaseemM.T., “QoS and Rate Enhancement in DVB-S2 using Fuzzy Rule Base System”, Journal of Intelligent & Fuzzy Systems (JIFS)30(1) (2016), 801-810.
26.
Atta-ur-Rahman, I.M.Qureshi, A.N.Malik, M.T.Naseem, “Dynamic Resource allocation for OFDM Systems using DE and Fuzzy Rule Base System”, Journal of Intelligent & Fuzzy Systems (JIFS)26(4) (2014), 2035-2046. doi:10.3233/IFS-130880.
27.
Atta-ur-Rahman, “Applications of Softcomputing in Adaptive Communication”, International Journal Control Theory and Applications10(18) (2017), 81-93,.
28.
Atta-ur-Rahman, I.M.Qureshi, A.N.Malik, “Adaptive Resource Allocation in OFDM Systems using GA and Fuzzy Rule Base System”, World Applied Sciences Journal (WASJ)18(6) (2012), 836-844. doi:10.5829/idosi.wasj.2012.18.06.906.
29.
Atta-ur-Rahman, I.M.Qureshi, A.N.Malik, “A Fuzzy Rule Base Assisted Adaptive Coding and Modulation Scheme for OFDM Systems”, J Basic Appl Sci Res2(5) (2012), 4843-4853.
30.
S.Das, B.K.Tripathy, Atta-ur-Rahman (Editors), Modeling, Analysis and Applications of Nature-Inspired Metaheuristic Algorithms, Edition: 1st, Publisher: IGI Global, (2017).
31.
Atta-ur-Rahman, I.M.Qureshi, “Optimum Resource Allocation in OFDM Systems using FRBS and Particle Swarm Optimization, “Nature and Biologically Inspired Computing (NaBIC’13), pp. 174-180, North Dakota, USA, (2013).
32.
Atta-ur-Rahman, I.M.Qureshi, M.Z.Muzaffar, M.T.Naseem, “A Fuzzy Rule Base Aided Rate Enhancement Scheme for OFDM Systems”. IEEE Conference on Emerging Technologies (ICET’12), pp. 151-156, October 8-9, 2012, Islamabad, Pakistan.
