Abstract
A differential evolution based methodology is introduced for the solution of elliptic partial differential equations (PDEs) with Dirichlet and/or Neumann boundary conditions. The solutions evolve over bounded domains throughout the interior nodes by minimization of nodal deviations among the population. The elliptic PDEs are replaced by the corresponding system of finite difference approximation, yielding an expression for nodal residues. The global residue is declared as the root-mean-square value of the nodal residues and taken as the cost function. The standard differential evolution is then used for the solution of elliptic PDEs by conversion to a minimization problem of the global residue. A set of benchmark problems consisting of both linear and nonlinear elliptic PDEs has been considered for validation, proving the effectiveness of the proposed algorithm. To demonstrate its robustness, sensitivity analysis has been carried out for various differential evolution operators and parameters. Comparison of the differential evolution based computed nodal values with the corresponding data obtained using the exact analytical expressions shows the accuracy and convergence of the proposed methodology.
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References
Ahmad I, Raja MAZ, Bilal M, et al., 2016. Bio-inspired computational heuristics to study Lane-Emden systems arising in astrophysics model. Springer Plus, 5, Article 1866. https://doi.org/10.1186/s40064-016-3517-2
Ahmad I, Raja MAZ, Bilal M, et al., 2017. Neural network methods to solve the Lane-Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neur Comput Appl, 28(S1):929–944. https://doi.org/10.1007/s00521-016-2400-y
Bramble JH, Hubbard BE, Thomée V, 1969. Convergence estimates for essentially positive type discrete Dirichlet problems. Math Comput, 23(108):695–709.
Carlson HA, Verberg R, Harris CA, 2017. Aeroservoelastic modeling with proper orthogonal decomposition. Phys Fluids, 29(2):020711. https://doi.org/10.1063/1.4975673
Carotenuto A, Ciccolella M, Massarotti N, et al., 2016. Models for thermo-fluid dynamic phenomena in low enthalpy geothermal energy systems: a review. Renew Sustain Energy Rev, 60:330–355. https://doi.org/10.1016/j.rser.2016.01.096
Chang HH, Tsai CY, 2014. Adaptive registration of magnetic resonance images based on a viscous fluid model. Comput Methods Progr Biomed, 117(2): 80–91. https://doi.org/10.1016/j.cmpb.2014.08.004
Courant R, Friedrichs K, Lewy H, 1928. Über die partiellen differenzengleichungen der mathematischen physik. Math Ann, 100(1):32–74 (in German). https://doi.org/10.1007/BF01448839
Dai CQ, Chen RP, Wang YY, et al., 2017a. Dynamics of light bullets in inhomogeneous cubic-quintic-septimal nonlinear media with PT-symmetric potentials. Nonl Dynam, 87(3):1675–1683. https://doi.org/10.1007/s11071-016-3143-0
Dai CQ, Liu J, Fan Y, et al., 2017b. Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality. Nonl Dynam, 88(2):1373–1383. https://doi.org/10.1007/s11071-016-3316-x
Das R, Singh K, Gogoi TK, 2017. Estimation of critical dimensions for a trapezoidal-shaped steel fin using hybrid diffierential evolution algorithm. Neur Comput Appl, 28(7):1683–1693. https://doi.org/10.1007/s00521-015-2155-x
Ding DJ, Jin DQ, Dai CQ, 2017. Analytical solutions of differential-difference sine-Gordon equation. Therm Sci, 21(4):1701–1705. https://doi.org/10.2298/TSCI160809056D
Fateh MF, Zameer A, Mirza NM, et al., 2017. Biologically inspired computing framework for solving two-point boundary value problems using differential evolution. Neur Comput Appl, 28(8): 2165–2179.
Gao LN, Zi YY, Yin YH, et al., 2017. Bäcklund transformation, multiple wave solutions and lump solutions to a (3+1)-dimensional nonlinear evolution equation. Nonl Dynam, 89(3): 2233–2240. https://doi.org/10.1007/s11071-017-3581-3
Gerschgorin VS, 1930. Fehlerabschätzung für das differenzenverfahren zur Lösung partieller differentialgleichungen. J Appl Math Mech, 10(4): 373–382 (in German). https://doi.org/10.1002/zamm.19300100409
Gilbarg D, Trudinger NS, 2015. Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Germany.
Guo W, Xu T, Lu ZL, 2016. An integrated chaotic time series prediction model based on efficient extreme learning machine and differential evolution. Neur Comput Appl, 27(4):883–898. https://doi.org/10.1007/s00521-015-1903-2
Hua YF, Guo BL, Ma WX, et al., 2019. Interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation for nonlinear waves. Appl Math Model, 74:184–198. https://doi.org/10.1016/j.apm.2019.04.044
Li YY, Zhao Y, Xie GN, et al., 2014. Local fractional Poisson and Laplace equations with applications to electrostatics in fractal domain. Adv Math Phys, 2014:590574.
Machado MR, Pereira VS, Costa DI, et al., 2016. Dynamics analysis of 1D structure including random parameter via frequency-domain state-vector equations. Proc 37th Iberian Latin-American Congress on Computational Mehods in Engineering, p.1–14.
Marichal Y, Chatelain P, Winckelmans G, 2014. An immersed interface solver for the 2-D unbounded Poisson equation and its application to potential flow. Comput Fluids, 96:76–86. https://doi.org/10.1016/j.compfluid.2014.03.012
Motzkin TS, Wasow W, 1952. On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J Math Phys, 31(1-4):253–259. https://doi.org/10.1002/sapm1952311253
Oden JT, Lima EA, Almeida RC, et al., 2016. Toward predictive multiscale modeling of vascular tumor growth. Arch Comput Methods Eng, 23(4):735–779. https://doi.org/10.1007/s11831-015-9156-x
Peaceman DW, Rachford HHJr, 1955. The numerical solution of parabolic and elliptic differential equations. J Soc Ind Appl Math, 3(1):28–41. https://doi.org/10.1137/0103003
Raja MAZ, Khan MAR, Mahmood T, et al., 2016a. Design of bio-inspired computing technique for nanofluidics based on nonlinear Jeffery-Hamel flow equations. Can J Phys, 94(5):474–489. https://doi.org/10.1139/cjp-2015-0440
Raja MAZ, Farooq U, Chaudhary NI, et al., 2016b. Stochastic numerical solver for nanofuidic problems containing multi-walled carbon nanotubes. Appl Soft Comput, 38:561–586. https://doi.org/10.1016/j.asoc.2015.10.015
Raja MAZ, Shah FH, Alaidarous ES, et al., 2017a. Design of bio-inspired heuristic technique integrated with interior-point algorithm to analyze the dynamics of heartbeat model. Appl Soft Comput, 52:605–629. https://doi.org/10.1016/j.asoc.2016.10.009
Raja MAZ, Ahmad I, Khan I, et al., 2017b. Neuro-heuristic computational intelligence for solving nonlinear pantograph systems. Front Inform Technol Electron Eng, 18(4):464–484. https://doi.org/10.1631/FITEE.1500393
Raja MAZ, Aslam MS, Chaudhary NI, et al., 2018. Bio-inspired heuristics hybrid with interior-point method for active noise control systems without identification of secondary path. Front Inform Technol Electron Eng, 19(2):246–259. https://doi.org/10.1631/FITEE.1601028
Selvadurai AP, 2000. Partial Differential Equations in Mechanics 2: the Biharmonic Equation, Poissona Equation. Springer Science & Business Media, Berlin, Germany.
Shao YL, Faltinsen OM, 2014. A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics. J Comput Phys, 274:312–332. https://doi.org/10.1016/j.jcp.2014.06.021
Spiller C, Toro EF, Vázquez-Cendón ME, et al., 2017. On the exact solution of the Riemann problem for blood flow in human veins, including collapse. Appl Math Comput, 303:178–179. https://doi.org/10.1016/j.amc.2017.01.024
Storn R, Price K, 1997. Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim, 11(4):341–359. https://doi.org/10.1023/A:1008202821328
Taleei A, Dehghan M, 2014. Direct meshless local Petrov-Galerkin method for elliptic interface problems with applications in electrostatic and elastostatic. Comput Methods Appl Mech Eng, 278:479–498. https://doi.org/10.1016/j.cma.2014.05.016
Vétois J, 2016. A priori estimates and application to the symmetry of solutions for critical p-Laplace equations. J Differ Equat, 260(1):149–161. https://doi.org/10.1016/j.jde.2015.08.041
Wang YY, Zhang YP, Dai CQ, 2016. Re-study on localized structures based on variable separation solutions from the modified tanh-function method. Nonl Dynam, 83(3):1331–1339. https://doi.org/10.1007/s11071-015-2406-5
Wang YY, Chen L, Dai CQ, et al., 2017. Exact vector multipole and vortex solitons in the media with spatially modulated cubica-quintic nonlinearity. Nonl Dynam, 90(2):1269–1275. https://doi.org/10.1007/s11071-017-3725-5
Webster AG, 2016. Partial Differential Equations of Mathematical Physics. Courier Dover Publications, Inc., New York, USA.
Yin YH, Ma WX, Liu JG, et al., 2018. Diversity of exact solutions to a (3+1)-dimensional nonlinear evolution equation and its reduction. Comput Math Appl, 76(6):1275–1283. https://doi.org/10.1016/j.camwa.2018.06.020
Zhang B, Zhang XL, Dai CQ, 2017. Discussions on localized structures based on equivalent solution with different forms of breaking soliton model. Nonl Dynam, 87(4):2385–2393. https://doi.org/10.1007/s11071-016-3197-z
Zhao LG, Liu HD, Zhao FK, 2013. Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. J Differ Equat, 255(1):1–23. https://doi.org/10.1016/j.jde.2013.03.005
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Muhammad Faisal FATEH, Aneela ZAMEER, Sikander M. MIRZA, Nasir M. MIRZA, Muhammad Saeed ASLAM, and Muhammad Asif Zahoor RAJA declare that they have no conflict of interest.
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Fateh, M.F., Zameer, A., Mirza, S.M. et al. Differential evolution based computation intelligence solver for elliptic partial differential equations. Frontiers Inf Technol Electronic Eng 20, 1445–1456 (2019). https://doi.org/10.1631/FITEE.1900221
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DOI: https://doi.org/10.1631/FITEE.1900221
Keywords
- Differential evolution
- Boundary value problems
- Partial differential equation
- Finite difference scheme
- Numerical computing