A simple spatial integration scheme for solving Cauchy problems of non-linear evolution equations. (English) Zbl 1398.65233
Summary: In this study, we address a new and simple non-iterative method to solve Cauchy problems of non-linear evolution equations without initial data. To start with, these ill-posed problems are analysed by utilizing a semi-discretization numerical scheme. Then, the resulting ordinary differential equations at the discretized times are numerically integrated towards the spatial direction by the group-preserving scheme (GPS). After that, we apply a two-stage GPS to integrate the semi-discretized equations. We reveal that the accuracy and stability of the new approach is very good from several numerical experiments even under a large random noisy effect and a very large time span.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
Keywords:
non-linear evolution equation; severely ill-posed problem; Cauchy problem; group preserving scheme (GPS); without initial conditionReferences:
| [1] | Satsuma J, In: Ablowitz M, Fuchssteiner B, Kruskal M, editors. Topics in soliton theory and exactly solvable nonlinear equations. Singapore: World Scientific; 1987. · Zbl 0721.00016 |
| [2] | Wang XY, Zhu ZS, Lu YK. Solitary wave solutions of the generalized Burgers-Huxley equation. J Phys A Math Gen. 1990;23:271-274.10.1088/0305-4470/23/3/011 · Zbl 0708.35079 |
| [3] | Wazwaz A-M. Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations. Appl Math Comput. 2008;195:754-761. · Zbl 1132.65098 |
| [4] | Deng X. Travelling wave solutions for the generalized Burgers-Huxley equation. Appl Math Comput. 2008;204:733-737. · Zbl 1160.35515 |
| [5] | Babolian E, Saeidian J. Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations. Commun Nonlinear Sci Numer Simul. 2009;14:1984-1992.10.1016/j.cnsns.2008.07.019 · Zbl 1221.65271 |
| [6] | Molabahrami A, Khani F. The homotopy analysis method to solve the Burgers-Huxley equation. Nonlinear Anal Real World Appl. 2009;10:589-600.10.1016/j.nonrwa.2007.10.014 · Zbl 1167.35483 |
| [7] | Deng XJ, Yan ZZ, Han L-B. Travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order. Chin Phys B. 2009;18:3169-3173. |
| [8] | Gao H, Zhao R-X. New exact solutions to the generalized Burgers-Huxley equation. Appl Math Comput. 2010;217:1598-1603. · Zbl 1202.35220 |
| [9] | Zhou Y, Liu Q, Zhang W. Bounded traveling waves of the generalized Burgers-Huxley equation. Nonlinear Anal Theory Methods Appl. 2011;74:1047-1060.10.1016/j.na.2010.09.012 · Zbl 1207.35052 |
| [10] | Krisnangkura M, Chinviriyasit S, Chinviriyasit W. Analytic study of the generalized Burger’s-Huxley equation by hyperbolic tangent method. Appl Math Comput. 2012;218:10843-10847. · Zbl 1278.35216 |
| [11] | Chen H, Zhang H. New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation. Chaos Solitons Fractals. 2004;19:71-76.10.1016/S0960-0779(03)00081-X · Zbl 1068.35126 |
| [12] | Ismail HNA, Raslan K, Abd Rabboh AA. Adomian decomposition method for Burger’s-Huxley and Burger’s-Fisher equations. Appl Math Comput. 2004;159:291-301. · Zbl 1062.65110 |
| [13] | Wazwaz A-M. The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations. Appl Math Comput. 2005;169:321-338. · Zbl 1121.65359 |
| [14] | Wazwaz A-M. Traveling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations. Appl Math Comput. 2005;169:639-656. · Zbl 1078.35109 |
| [15] | El-Wakil SA, Abdou MA. Modified extended tanh-function method for solving nonlinear partial differential equations. Chaos Solitons Fractals. 2007;31:1256-1264.10.1016/j.chaos.2005.10.072 · Zbl 1139.35389 |
| [16] | Abbasbandy S. Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method. Appl Math Model. 2008;32:2706-2714.10.1016/j.apm.2007.09.019 · Zbl 1167.35395 |
| [17] | Abdusalam HA. Analytic and approximate solutions for Nagumo telegraph reaction diffusion equation. Appl Math Comput. 2004;157:515-522. · Zbl 1054.65104 |
| [18] | Aronson DG, Weinberger HF. Multidimensional nonlinear diffusion arising in population genetics. Adv Math. 1978;30:33-76.10.1016/0001-8708(78)90130-5 · Zbl 0407.92014 |
| [19] | Browne P, Momoniat E, Mahomed FM. A generalized Fitzhugh-Nagumo equation. Nonlinear Anal. 2008;68:1006-1015.10.1016/j.na.2006.12.001 · Zbl 1135.35009 |
| [20] | Kawahara T, Tanaka M. Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation. Phys Lett A. 1983;97:311-314.10.1016/0375-9601(83)90648-5 |
| [21] | Li H, Guo Y. New exact solutions to the Fitzhugh-Nagumo equation. Appl Math Comput. 2006;180:524-528. · Zbl 1102.35315 |
| [22] | Aasaraai A. Analytic solution for Newell-Whitehead-Segel equation by differential transform method. Middle-East J Sci Res. 2011;10:270-273. |
| [23] | Batiha B, Noorani MSM, Hashim I. Numerical simulation of the generalized Huxley equation by He’s variational iteration method. Appl Math Comput. 2007;186:1322-1325. · Zbl 1118.65367 |
| [24] | Hashemi SH, Mohammadi Daniali HR, Ganji DD. Numerical simulation of the generalized Huxley equation by He’s homotopy perturbation method. Appl Math Comput. 2007;192:157-161. · Zbl 1193.65181 |
| [25] | Darvishi MT, Kheybari S, Khani F. Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burgers-Huxley equation. Commun Nonlinear Sci Numer Simul. 2008;13:2091-2103.10.1016/j.cnsns.2007.05.023 · Zbl 1221.65261 |
| [26] | Khattak AJ. A computational meshless method for the generalized Burger’s-Huxley equation. Appl Math Model. 2009;33:3718-3729.10.1016/j.apm.2008.12.010 · Zbl 1185.65191 |
| [27] | Macías-Díaz JE, Ruiz-Ramírez J, Villa J. The numerical solution of a generalized Burgers-Huxley equation through a conditionally bounded and symmetry-preserving method. Comput Math Appl. 2011;61:3330-3342.10.1016/j.camwa.2011.04.022 · Zbl 1222.65095 |
| [28] | Zhang R, Yu X, Zhao G. The local discontinuous Galerkin method for Burger’s-Huxley and Burger’s-Fisher equations. Appl Math Comput. 2012;218:8773-8778. · Zbl 1245.65130 |
| [29] | Çelik İ. Haar wavelet method for solving generalized Burger’s-Huxley equation. Arab J Math Sci. 2012;18:25-37. · Zbl 1236.65130 |
| [30] | Dehghan M, Saray BN, Lakestani M. Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers-Huxley equation. Math Comput Model. 2012;55:1129-1142. · Zbl 1255.65182 |
| [31] | Ervin VJ, Macías-Díaz JE, Ruiz-Ramírez J. A positive and bounded finite element approximation of the generalized Burgers-Huxley equation. J Math Anal Appl. 2015;424:1143-1160.10.1016/j.jmaa.2014.11.047 · Zbl 1306.65263 |
| [32] | Duan Y, Kong L, Zhang R. A lattice Boltzmann model for the generalized Burgers-Huxley equation. Phys A. 2012;391:625-632.10.1016/j.physa.2011.08.034 |
| [33] | Kaya D, El-Sayed SM. A numerical simulation and explicit solutions of the generalized Burgers-Fisher equation. Appl Math Comput. 2004;152:403-413. · Zbl 1052.65098 |
| [34] | Zhu CG, Kang WS. Numerical solution of Burgers-Fisher equation by cubic B-spline quasi-interpolation. Appl Math Comput. 2010;216:2679-2686. · Zbl 1193.65177 |
| [35] | Javidi M. A modified Chebyshev pseudospectral DD algorithm for the GBH equation. Comput Math Appl. 2011;62:3366-3377.10.1016/j.camwa.2011.08.051 · Zbl 1236.65122 |
| [36] | Soheili AR, Kerayechian A, Davoodi N. Adaptive numerical method for Burgers-type nonlinear equations. Appl Math Comput. 2012;219:3486-3495. · Zbl 1311.65113 |
| [37] | Zhao T, Li C, Zang Z, et al. Chebyshev-Legendre pseudo-spectral method for the generalized Burgers-Fisher equation. Appl Math Model. 2012;36:1046-1056.10.1016/j.apm.2011.07.059 · Zbl 1243.65126 |
| [38] | Xiang X, Wang Z, Shi B. Modified lattice Boltzmann scheme for nonlinear convection diffusion equations. Commun Nonlinear Sci Numer Simul. 2012;17:2415-2425.10.1016/j.cnsns.2011.09.036 · Zbl 1335.76044 |
| [39] | Bhrawy AH. A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients. Appl Math Comput. 2013;222:255-264. · Zbl 1329.65234 |
| [40] | Jiwari R, Gupta RK, Kumar V. Polynomial differential quadrature method for numerical solutions of the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. Ain Shams Eng J. 2014;5:1343-1350.10.1016/j.asej.2014.06.005 |
| [41] | Nourazar SS, Soori M, Nazari-Golshan A. On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method. Aust J Basic Appl Sci. 2011;5:1400-1411. |
| [42] | Ezzati R, Shakibi K. Using Adomian’s decomposition and multi-quadric quasi-interpolation methods for solving Newell-Whitehead equation. Proc Comput Sci. 2011;3:1043-1048.10.1016/j.procs.2010.12.171 |
| [43] | Macías-Díaz JE, Ruiz-Ramírez J. A non-standard symmetry-preserving method to compute bounded solutions of a generalized Newell-Whitehead-Segel equation. Appl Numer Math. 2011;61:630-640.10.1016/j.apnum.2010.12.008 · Zbl 1366.65079 |
| [44] | Pue-on P. Laplace adomian decomposition method for solving Newell-Whitehead-Segel equation. Appl Math Sci. 2013;7:6593-6600.10.12988/ams.2013.310603 |
| [45] | Saravanan A, Magesh N. A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation. J Egypt Math Soc. 2013;21:259-265.10.1016/j.joems.2013.03.004 · Zbl 1281.65134 |
| [46] | Jassim HK. Homotopy perturbation algorithm using Laplace transform for Newell-Whitehead-Segel equation. Int J Adv Appl Math Mech. 2015;2:8-12. · Zbl 1359.65226 |
| [47] | Aminikhah H, Alavi J. Numerical study of the nonlinear Cauchy diffusion problem and Newell-Whitehead equation via cubic B-spline quasi-interpolation. Iran J Numer Anal Optim. 2015;5:63-72. · Zbl 1330.65128 |
| [48] | Liu C-S. Cone of non-linear dynamical system and group preserving schemes. Int J Non-Linear Mech. 2001;36:1047-1068.10.1016/S0020-7462(00)00069-X · Zbl 1243.65084 |
| [49] | Lee HC, Liu C-S. The fourth-order group preserving methods for the integrations of ordinary differential equations. Comput Model Eng Sci. 2009;41:1-5.10.1063/1.3160130 · Zbl 1357.65088 |
| [50] | Liu C-S. Nonstandard group-preserving schemes for very stiff ordinary differential equations. Comput Model Eng Sci. 2005;9:255-272. · Zbl 1357.65090 |
| [51] | Liu C-S. Preserving constraints of differential equations by numerical methods based on integrating factors. Comput Model Eng Sci. 2006;12:83-107. · Zbl 1232.65137 |
| [52] | Chen Y-W, Liu C-S, Chang J-R. A chaos detectable and time step-size adaptive numerical scheme for non-linear dynamical systems. J Sound Vibr. 2007;299:977-989.10.1016/j.jsv.2006.08.028 · Zbl 1243.65082 |
| [53] | Liu C-S. An efficient backward group preserving scheme for the backward in time Burgers equation. Comput Model Eng Sci. 2006;12:55-65. · Zbl 1232.65130 |
| [54] | Liu C-S. A group preserving scheme for Burgers equation with very large Reynolds number. Comput Model Eng Sci. 2006;12:197-211. · Zbl 1232.76012 |
| [55] | Liu C-S. The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions. Comput Model Eng Sci. 2006;13:149-163. · Zbl 1232.65108 |
| [56] | Liu C-S. Efficient shooting methods for the second order ordinary differential equations. Comput Model Eng Sci. 2006;15:69-86. · Zbl 1152.65453 |
| [57] | Liu C-S. The Lie-group shooting method for singularly perturbed two-point boundary value problems. Comput Model Eng Sci. 2006;15:179-196. · Zbl 1152.65452 |
| [58] | Liu C-S. The computations of large rotation through an index two nilpotent matrix. Comput Model Eng Sci. 2006;16:157-175. |
| [59] | Liu C-S. New integrating methods for time-varying linear systems and Lie-group computations. Comput Model Eng Sci. 2007;20:157-175. · Zbl 1152.93340 |
| [60] | Liu C-S, Chang C-W, Chang J-R. Past cone dynamics and backward group preserving schemes for backward heat conduction problems. Comput Model Eng Sci. 2006;12:67-81. · Zbl 1232.65129 |
| [61] | Liu C-S, Chang J-R, Chang C-W. The Lie-group shooting method for steady-state Burgers equation with high Reynolds number. J Hydrodyn Ser B. 2006;18:367-372.10.1016/S1001-6058(06)60080-2 |
| [62] | Liu C-S, Chang C-W, Chang J-R. A new shooting method for solving boundary layer equations in fluid mechanics. Comput Model Eng Sci. 2008;32:1-15. · Zbl 1232.65104 |
| [63] | Liu C-S, Chang C-W, Chang J-R. The backward group preserving scheme for 1D backward in time advection-dispersion equation. Numer Methods Partial Differ Equ. 2010;26:61-80.10.1002/num.v26:1 · Zbl 1425.65100 |
| [64] | Chang C-W, Liu C-S, Chang J-R. A group preserving scheme for inverse heat conduction problems. Comput Model Eng Sci. 2015;10:13-38. · Zbl 1232.80005 |
| [65] | Chang C-W, Chang J-R, Liu C-S. The Lie-group shooting method for boundary layer equations in fluid mechanics. J Hydrodyn Ser B. 2006;18:103-108.10.1016/S1001-6058(06)60038-3 |
| [66] | Chang J-R, Liu C-S, Chang C-W. A new shooting method for quasi-boundary regularization of backward heat conduction problems. Int J Heat Mass Transfer. 2007;50:2325-2332.10.1016/j.ijheatmasstransfer.2006.10.050 · Zbl 1123.80005 |
| [67] | Chang C-W, Liu C-S, Chang J-R. The Lie-group shooting method for quasi-boundary regularization of backward heat conduction problems. ICCES Online J. 2007;3:69-80. |
| [68] | Chang C-W, Chang J-R, Liu C-S. The Lie-group shooting method for solving classical Blasius flat-plate problem. Comput Mater Cont. 2008;7:139-153. · Zbl 1231.76082 |
| [69] | Chang C-W, Liu C-S, Chang J-R. A new shooting method for quasi-boundary regularization of multi-dimensional backward heat conduction problems. J Chin Inst Eng. 2009;32:307-318.10.1080/02533839.2009.9671510 |
| [70] | Liu C-S, Chang C-W. A Lie-group adaptive method to identify the radiative coefficients in parabolic partial differential equations. Comput Mater Cont. 2011;25:107-134. |
| [71] | Liu C-S, Chang C-W. A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains. Eng Anal Boundary Elem. 2012;36:211-219.10.1016/j.enganabound.2011.08.001 · Zbl 1245.65151 |
| [72] | Liu C-S, Chang C-W. A simple algorithm for solving Cauchy problem of nonlinear heat equation without initial value. Int J Heat Mass Transfer. 2015;80:562-569.10.1016/j.ijheatmasstransfer.2014.09.053 |
| [73] | Hariharan G, Kannan K, Sharma KR. Haar wavelet method for solving Fisher’s equation. Appl Math Comput. 2009;211:284-292. · Zbl 1162.65394 |
| [74] | Arnăutu V, Morosanu C. Numerical approximation for the phase-field transition system. Int J Comput Math. 1996;62:209-221.10.1080/00207169608804538 · Zbl 0867.35104 |
| [75] | Morosanu C. Approximation of the phase-field transition system via fractional steps method. Numer Funct Anal Optim. 1997;18:623-648.10.1080/01630569708816782 · Zbl 0909.35008 |
| [76] | Morosanu C, Motreanu D. Uniqueness and approximation for the nonlinear parabolic equation in Caginalp’s model. Int J Appl Math. 2000;2:113-129. · Zbl 1171.35402 |
| [77] | Benincasa T, Morosanu C. Fractional steps scheme to approximate the phase-field transition system with nonhomogeneous Cauchy-Neumann boundary conditions. Numer Funct Anal Optim. 2009;30:199-213.10.1080/01630560902841120 · Zbl 1177.35019 |
| [78] | Benincasa T, Morosanu C. A product formula approach to an inverse problem governed by nonlinear phase-field transition system. Case 1D. ROMAI J. 2010;6:15-26. · Zbl 1313.35147 |
| [79] | Morosanu C. Cubic spline method and fractional steps schemes to approximate the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions. ROMAI J. 2012;8:73-91. · Zbl 1313.76075 |
| [80] | Morosanu C, Moşneagu A-M. On the numerical approximation of the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions. Case 1D. ROMAI J. 2013;9:91-110. · Zbl 1299.65195 |
| [81] | Morosanu C, Gheorghiu CI. Accurate spectral solutions to a phase-field transition system. ROMAI J. 2014;10:89-99. · Zbl 1340.65236 |
| [82] | Miranville A, Morosanu C. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Dis Cont Dyn Syst - Ser S. 2016;9:537-556. 10.3939/dcdss.2016011 · Zbl 1353.65094 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
