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Review of wavelet methods for the solution of reaction-diffusion problems in science and engineering. (English) Zbl 1427.65429

Summary: Wavelet method is a recently developed tool in applied mathematics. Investigation of various wavelet methods, for its capability of analyzing various dynamic phenomena through waves gained more and more attention in engineering research. Starting from ’offering good solution to differential equations’ to capturing the nonlinearity in the data distribution, wavelets are used as appropriate tools at various places to provide good mathematical model for scientific phenomena, which are usually modeled through linear or nonlinear differential equations. Review shows that the wavelet method is efficient and powerful in solving wide class of linear and nonlinear reaction-diffusion equations. This review intends to provide the great utility of wavelets to science and engineering problems which owes its origin to 1919. Also, future scope and directions involved in developing wavelet algorithm for solving reaction-diffusion equations are addressed.

MSC:

65T60 Numerical methods for wavelets
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis

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[1] Monsour, M. B.A., Travelling wave solutions of a nonlinear reaction-diffusion-chemotaxis model for bacterial pattern formation, Appl. Math. Model., 32, 240-247 (2008) · Zbl 1134.35067
[2] Carey, G.; Fowkes, N.; Staelens, A.; Pardhanani, A., A class of coupled nonlinear reaction diffusion models exhibiting fingering, J. Comput. Appl. Math., 166, 87-99 (2004) · Zbl 1107.76418
[3] Kuramoto, Y., Chemical Oscillations, Waves and Turbulence (1984), Springer: Springer Berlin · Zbl 0558.76051
[4] Hariharan, G.; Ponnusamy, V.; Srikanth, R., Wavelet method to film-pore diffusion model for methylene blue adsorption onto plant leaf powders, J. Math. Chem., 50, 2775-2785 (2012) · Zbl 1308.74102
[6] Oran, S.; Boris, J. P., Numerical Simulation of Reactive Flow (1987), Elsevier: Elsevier New York · Zbl 0875.76678
[7] Allen, S. M.; Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27, 1085 (1979)
[8] Kolmogorov, A.; Petrovsky, I.; Piskunov, N., Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Moscow Bull. Univ. Math., A1, 1-26 (1937) · Zbl 0018.32106
[9] Zhou, M. C., An application of traveling wave analysis in economic growth model, Appl. Math. Comput., 200, 261-266 (2008) · Zbl 1136.91529
[10] Sherratt, J., On the transition from initial data traveling waves in the Fisher-KPP equation, Dyn. Stab. Syst., 13, 2, 167-174 (1998) · Zbl 0912.35084
[11] Aronson, D. G.; Weinberger, H. F., Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30, 33-76 (1978) · Zbl 0407.92014
[14] Rosenau, P.; Hyman, J. M., Compactons: solitons with finite wavelengths, Phys. Rev. Lett., 70, 5, 564-567 (1993) · Zbl 0952.35502
[15] Murray, J. D., Mathematical Biology (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0779.92001
[16] Liao, W.; Zhu, J.; Khaliq, A. Q.M., An efficient high order algorithm for solving systems of reaction-diffusion equations, J. Numer. Methods Partial Di’erential Equ., 18, 340-354 (2002) · Zbl 0997.65105
[17] Abbasbandy, S., Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Appl. Math. Model., 32, 2706-2714 (2008) · Zbl 1167.35395
[18] Abdusalam, H. A., Analytic and approximate solutions for Nagumo telegraph reaction diffusion equation, Appl. Math. Comput., 157, 515-522 (2004) · Zbl 1054.65104
[19] Ablowitz, M.; Zepetella, A., Explicit solutions of Fisher’s equation for a special wave speed, Bull. Math. Biol., 41, 835-840 (1979) · Zbl 0423.35079
[20] Al-Khaled, K., Numerical study of Fisher’s reaction-diffusion equation by the Sinc collocation method, J. Comput. Appl. Math., 137, 245-255 (2001) · Zbl 0992.65108
[21] Baronas, R.; Ivanauskas, F.; Kulys, J., Modeling dynamics of amperometric biosensors in batch and flow injection analysis, J. Math. Chem., 32, 2, 225-237 (2002) · Zbl 1011.92502
[22] Batiha, B.; Noorani, M. S.M.; Hashim, I., Numerical simulation of the generalized Huxley equation by He’s variational iteration method, Appl. Math. Comput., 186, 2, 1322-1325 (2007) · Zbl 1118.65367
[23] Batiha, B.; Noorani, M. S.M.; Hashim, I., Application of variational iteration method to the generalized Burgers-Huxley equation, Chaos Solitons Fractals, 36, 3, 660-663 (2008) · Zbl 1141.49006
[24] Burger, R.; Ruiz-Baier, R., Multiresolution simulation of reaction-diffusion systems with strong degeneracy, Bol. Soc. Esp. Mat. Apl., 47, 73-80 (2009) · Zbl 1242.65176
[25] Ismail, H. N.A.; Raslan, K.; Abd Rabboh, A. A., Adomian decomposition method for Burger’s-Huxley and Burger’s-Fisher equations, Appl. Math. Comput., 159, 1, 291-301 (2004) · Zbl 1062.65110
[26] Javidi, M., A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method, Appl. Math. Comput., 178, 2, 338-344 (2006) · Zbl 1100.65081
[27] Molabahramia, A.; Khani, F., The homotopy analysis method to solve the Burgers-Huxley equation, Nonlinear Anal. Real World Appl., 10, 2, 589-600 (2009) · Zbl 1167.35483
[28] Olmos, D.; Shizgal, Bernie D., Pseudospectral method of solution of Fisher’s equation, J. Comput. Appl. Math., 193, 219-242 (2006) · Zbl 1092.65088
[29] Rajendran, L.; Senthamarai, R., Traveling-wave solution of non-linear coupled reaction-diffusion equation arising in mathematical chemistry, J. Math. Chem., 46, 550-561 (2009) · Zbl 1223.92078
[30] Ramos, J. I., A finite volume method for one-dimensional reaction-diffusion problems, Appl. Math. Comput., 188, 739-748 (2007) · Zbl 1117.65128
[31] Ramos, J. I., Implicit, compact, linearized \(-methods with factorization for multidimensional reaction-diffusion equations, Appl. Math. Comput., 94, 17-43 (1998\) · Zbl 0943.65098
[32] Chui, C. K.; Quak, E., Wavelets on a bounded interval, (Braess, D.; Schumaker, L. L., Numerical Methods of Approximation Theory (1992), Birkhauser: Birkhauser Basel), 1-24
[34] Cohen, A., Wavelets in numerical analysis, (Ciarlet, P. G.; Lions, J. L., The Handbook of Numerical Analysis, vol. VII (1999), Elsevier: Elsevier Amsterdam)
[35] Dahmen, W., Wavelet methods for PDEs/some recent developments, J. Comput. Appl. Math., 128, 133-185 (2001) · Zbl 0974.65101
[36] Strang, Gilbert, Wavelet transforms versus Fourier transforms, Bull. Am. Math. Soc., 28, 2, 288-305 (1993) · Zbl 0771.42021
[37] Goswami, J. C., Chan, Fundamentals of Wavelets. Theory, Algorithms, and Applications (1999), John Wiley and Sons: John Wiley and Sons New York · Zbl 1209.65156
[38] Kumar, Manoj; Pandit, Sapna, Wavelet transform and wavelet based numerical methods: an introduction, Int. J. Nonlinear Sci., 13, 3, 325-345 (2012) · Zbl 1394.65176
[39] Hariharan, G.; Kannan, K.; Sharma, K. R., Haar wavelet method for solving Fisher’s equation, Appl. Math. Comput., 211, 284-292 (2009) · Zbl 1162.65394
[40] Soman, K. P.; Ramachandran, K. I., Insight into Wavelet Transform From Theory to Practice (2004), Prentice-Hall of India private limited: Prentice-Hall of India private limited New Delhi
[42] Cattani, C., Connection coefficients of Shannon wavelets, Math. Model. Anal., 11, 2, 117-132 (2006) · Zbl 1117.65179
[43] Hariharan, G.; Kannan, K., A comparative study of a Haar wavelet method and a restrictive Taylor’s series method for solving convection-diffusion equations, Int. J. Comput. Methods Eng. Sci. Mech., 11, 4, 173-184 (2010) · Zbl 1230.65109
[44] Gu, J. S.; Jiang, W. S., The Haar wavelets operational matrix of integration, Int. J. Syst. Sci., 27, 7, 623-628 (1996) · Zbl 0875.93116
[45] Stankovi, R. S.; Falkowski, B. J., The Haar wavelet transform: its status and achievements, Comput. Electr. Eng., 29, 25-44 (2003) · Zbl 1059.65132
[47] Al-Bayati, A. Y.; Ibraheem, K. I.; Ghatheth, A. I., A modified wavelet algorithm to solve BVPs with an infinite number of boundary conditions, Int. J. Open Problems Comput. Math., 24, 2, 141-154 (2011)
[48] Javidi, M.; Golbabai, A., A new domain decomposition algorithm for generalized Burger’s-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Solitons Fractals, 39, 2, 849-857 (2009) · Zbl 1197.65153
[49] Hariharan, G., Wavelet solutions for a class of fractional Klein-Gordon equations, J. Comput. Nonlinear Dyn., 8, 021008-1 (2013)
[50] Beylkin, G., On the representation of operators in bases of compactly supported wavelets, SIAM J. Numer. Anal., 29, 507-537 (1993)
[51] Beylkin, G.; Keiser, J. M., An adaptive pseudo-wavelet approach for solving nonlinear partial differential equations, (Dahmen, W.; Kurdila, A. J.; Oswald, P., Multiscale Wavelet Methods for PDEs (1997), Academic Press: Academic Press New York), 137-197
[52] Daubechies, I., Ten Lectures on Wavelets (1992), SIAM: SIAM Philadelphia · Zbl 0776.42018
[53] Comincioli, V.; Naldi, G.; Scapolla, T., A wavelet-based method for numerical solution of nonlinear evolution equations, Appl. Numer. Math., 33, 291-297 (2000) · Zbl 0964.65112
[54] Cruz, P.; Mendes, A.; Magalhaes, F. D., Using wavelets for solving PDEs: an adaptive collocation method, Chem. Eng. Sci., 56, 3305-3309 (2001)
[55] Rathish Kumar, B. V.; Mehra, Mani, A time-accurate pseudo-wavelet scheme for parabolic and hyperbolic PDE’s, Nonlinear Anal., 63, e345-e356 (2005) · Zbl 1159.65372
[56] Rathish Kumar, B. V.; Mehra, Mani, A time accurate pseudo-wavelet scheme for two-dimensional turbulence, Int. J. Wavelets Multiresolution Inf. Process., 3, 4, 587-599 (2005) · Zbl 1085.76057
[57] Rathish Kumar, B. V.; Mehra, Mani, (Time Accurate Fast Wavelet-Taylor Galerkin Method for Partial Differential Equations (2009), Wiley InterScience), doi:10.1002/num.20092 · Zbl 1170.65331
[58] Barey, M.; Mallat, S.; Papanicolaou, G., A wavelet based space-time adaptive numerical method for partial differential equations, Math. Model Numer. Anal., 26, 703-734 (1992)
[59] Bertoluzza, S., A wavelet collocation method for the numerical solution of partial differential equations, Appl. Comput. Harmonic Anal., 3, 1-9 (1996) · Zbl 0853.65122
[60] Bertoluzza, S., Adaptive wavelet collocation method for the solution of Burgers equation, Transp. Theory Stat., 5, 339 (1996) · Zbl 0868.65071
[61] Chen, C. F.; Hsiao, C. H., Haar wavelet method for solving lumped and distributed-parameter systems, IEEE Proc.: Part D, 144, 1, 87-94 (1997) · Zbl 0880.93014
[64] Cattani, C., Harmonic wavelet solutions of the Schrödinger equation, Int. J. Fluid Mech. Res., 30, 5, 463-472 (2003)
[65] Cattani, C.; Rushchitsky, J., Wavelet and wave analysis as applied to materials with micro or nanostructure, (Series on Advances in Mathematics for Applied Sciences, vol. 74 (2007), World Scientific Publishing: World Scientific Publishing Singapore) · Zbl 1152.74001
[67] Cattani, C., Harmonic wavelets towards the solution of nonlinear PDE, Comput. Math. Appl., 50, 8-9, 1191-1210 (2005) · Zbl 1118.65133
[68] Celik, I., Haar wavelet method for solving generalized Burgers-Huxley eq, Arab J. Math. Sci., 18, 25-37 (2011) · Zbl 1236.65130
[69] Chen, Y.; Wu, Y., Wavelet method for a class of fractional convection-diffusion equation with variable coefficients, J. Comput. Sci., 1, 146-149 (2010)
[70] Cruz, P.; Mendes, A.; Magalhaes, F. D., A wavelet based adaptive grid method for the solution of partial differential equations, Am. Inst. Chem. Eng. J., 48, 774-785 (2002)
[71] Ewing, R. E.; Liu, J.; Wang, H., Adaptive biorthogonal spline schemes for advection-reaction equations, J. Comput. Phys., 193, 21-39 (2003) · Zbl 1056.76065
[72] Mishra, S. K.; Muralidharan, K.; Pannala, S.; Simunovic, S.; Deymier, P.; Frantziskonis, G., Wavelet-based spatial scaling of coupled reaction-diffusion fields, Int. J. Multiscale Comput. Eng., 6, 281-297 (2008)
[73] Muralidharan, K.; Mishra, S. K.; Frantziskonis, G.; Deymier, P. A.; Nukala, P.; Simunovic, S.; Pannala, S., The dynamic compound wavelet matrix method for multiphysics/multiscale problems, Phys. Rev. E, 77, 2008, 026714 (2008)
[74] Frantziskonis, G.; Muralidharan, K.; Deymier, P.; Simunovic, S.; Nukala, P.; Pannala, S., Time-parallel multiscale/multiphysics framework, J. Comp. Phys., 228, 8085-8092 (2009) · Zbl 1175.65080
[75] Farge, M.; Kevlahan, N.; Perrier, V.; Goirand, E., Wavelets and turbulence, Proc. IEEE, 84, 4, 639-669 (1996)
[76] Gu, Y.; Liao, W.; Zhu, J., An efficient high-order algorithm for solving systems of 3-D reaction-diffusion equations, J. Comput. Appl. Math., 155, 1-17 (2003) · Zbl 1019.65065
[78] Hariharan, G.; Kannan, K., Kal Renganathan Sharma, Haar wavelet in estimating depth profile of soil temperature, Appl. Math. Comput., 210, 119-125 (2009) · Zbl 1173.80307
[79] Hariharan, G.; Kannan, K., Haar wavelet method for solving Cahn-Allen equation, Appl. Math. Sci., 3, 51, 2523-2533 (2009) · Zbl 1187.65113
[80] Hariharan, G.; Kannan, K., Haar wavelet method for solving FitzHugh-Nagumo equation, Intl. J. Math. Stat. Sci., 2, 2 (2010)
[81] Hariharan, G.; Kannan, K., A comparison of Haar wavelet and Adomain decomposition method for solving one-dimensional reaction-diffusion equations, Int. J. Appl. Math. Comput., 2, 1, 50-61 (2010)
[82] Hariharan, G., Wavelet Method for a Class of Fractional Klein-Gordon Equations, J. Comput. Nonlinear Dynam., 8, 2, 021008 (2013)
[83] Hariharan, G., Haar Wavelet method for solving sine-Gordon and Klein-Gordon equations, Int. J. Nonlinear Sci., 9, 2, 1-10 (2010)
[84] Hariharan, G.; Kannan, K., A comparative study of Haar wavelet method and homotopy perturbation method for solving one-dimensional reaction-diffusion equations, Int. J. Appl. Math. Comput., 3, 1, 21-34 (2011)
[85] Hariharan, G.; Kannan, K., Haar wavelet method for solving nonlinear parabolic equations, J. Math. Chem., 48, 1044-1061 (2010) · Zbl 1207.35183
[86] Lepik, U., Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul., 68, 127-143 (2005) · Zbl 1072.65102
[87] Lepik, U., Application of the Haar wavelet transform to solving integral and differential Equations, Proc. Estonian Acad. Sci. Phys. Math., 56, 1, 28-46 (2007) · Zbl 1143.65104
[88] Lepik, U., Numerical solution of evolution equations by the Haar wavelet method, J. Appl. Math. Comput., 185, 695-704 (2007) · Zbl 1110.65097
[89] Lepik, U., Solving PDEs with the aid of two-dimensional Haar wavelets, Comput. Math. Appl., 61, 1873-1879 (2011) · Zbl 1219.65169
[90] Lepik, Ű., Haar wavelet method for higher order differential equations, Int. J. Math. Comput., 1, 84-94 (2008)
[91] Li, Y.; Wang, D., Wavelet method for nonlinear partial differential equations of fractional order, Comput. Inf. Sci., 4, 5, 28-35 (2011)
[92] Mallat, S., Multiresolution approximation and wavelet orthogonal bases of L2ðRÞ, Trans. Am. Math. Soc., 315, 69-87 (1989) · Zbl 0686.42018
[93] Jiwari, Ram., Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation, Comput. Phys. Commun., 183, 11, 2413-2423 (2012) · Zbl 1302.35337
[94] Santos, J. C.; Cruz, P.; Alves, M. A.; Oliveira, P. J.; Magalhaes, F. D.; Mendes, A., Adaptive multiresolution approach for two-dimensional PDEs, Comput. Methods Appl. Mech. Eng., 193, 405-425 (2004) · Zbl 1075.65129
[95] Cai, Wei; Zhang, Wu, An adaptive spline wavelet ADI (SW-ADI) method for two-dimensional reaction-diffusion equations, J. Comp. Phys., 139, 92-126 (1998) · Zbl 0905.65103
[96] Schwab, C.; Stevenson, R., Adaptive wavelet algorithms for elliptic PDE’s on product domains, Math. Comp., 77, 71-92 (2008), (electronic) Zbl 1127.41009 · Zbl 1127.41009
[97] Sousa, J. M.; Cruz, P.; Magalhães, F. D.; Mendes, Adélio, Modeling catalytic membrane reactors using an adaptive wavelet-based collocation method, J. Membr. Sci., 208, 57-68 (2002)
[98] Alves, M. A.; Cruz, P.; Mendes, A.; Magalhaes, F. D.; Pinho, F. T.; Oliveira, P. J., Adaptive multiresolution approach for solution of hyperbolic PDEs, Comput. Methods Appl. Mech. Eng., 191, 3909-3928 (2002) · Zbl 1010.65042
[99] Holmstrom, M., Solving hyperbolic PDEs using interpolating wavelets, SIAM J. Sci. Comput., 21, 405-420 (1999) · Zbl 0959.65109
[100] Chen, M. Q.; Hwang, C.; Shih, Y. P., The computation of wavelet-Galerkin approximation on a bounded interval, Int. J. Numer. Methods Eng., 39, 2921-2944 (1996) · Zbl 0884.76058
[101] Amaratunga, K.; Williams, J.; Wavelet-Galerkin, R., Solutions for one dimensional partial differential equations, Int. J. Numer. Methods Eng., 37, 2703-2716 (1992) · Zbl 0813.65106
[102] Kumar, Vivek.; Mehra, Mani., Cubic spline adaptive wavelet scheme to solve singularly perturbed reaction-diffusion problems, Int. J. Wavelets Multiresolut. Inf. Process., 5, 2, 317-331 (2007) · Zbl 1153.65379
[103] Avudainayagam, A.; Vani, C., Wavelet-Galerkin solutions of quasilinear hyperbolic conservation equations, Commun. Numer., 15, 8, 589-601 (1999) · Zbl 0942.65114
[104] Slavova, A.; Zecca, P., CNN model for studying dynamics and travelling wave solutions of FitzHugh-Nagumo equation, J. Comput. Appl. Math., 151, 13-24 (2003) · Zbl 1049.92005
[105] Jahnke, Tobias., An adaptive wavelet method for the chemical master equation, SIAM J. Sci. Comput., 31, 6, 4373-4394 (2010) · Zbl 1205.65022
[106] Bindal, Aditya; Khinast, Johannes G.; Ierapetritou, Marianthi G., Adaptive multiscale solution of dynamical systems in chemical processes using wavelets, Comput. Chem. Eng., 27, 131-142 (2003)
[107] Stundzia, A.; Lumsden, C., Stochastic simulation of coupled reaction-diffusion processes, J. Comput. Phys., 127, 196-207 (1996) · Zbl 0860.65122
[108] Teixeira, J., Stable schemes for partial differential equations: the one-dimensional reaction-diffusion equation, Math. Comput. Simul., 64, 507-520 (2004) · Zbl 1039.65065
[109] Wazwaz, A. M.; Gorguis, A., An analytical study of Fisher’s equation by using Adomian decomposition method, Appl. Math. Comput., 154, 3, 609-620 (2004) · Zbl 1054.65107
[110] Majak, J.; Pohlak, M.; Eerme, M.; Lepikult, T., Weak formulation based Haar wavelet method for solving differential equations, Appl. Math. Comput., 211, 488-494 (2009) · Zbl 1162.65395
[112] Wang, M.; Zhao, F., Haar wavelet method for solving two-dimensional Burgers’ equation, Adv. Intell. Soft Comput., 145, 381-387 (2012)
[113] Siraj-ul-Islam, Imran; Aziz, Fazal; Haq, Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems, Int. J. Therm. Sci., 50, 5, 686-697 (2011) · Zbl 1238.65080
[114] Farge, M., Wavelet transforms and their applications to turbulence, Ann. Rev. Fluid Mech., 24, 395-457 (1992) · Zbl 0743.76042
[115] Whitham, G. B., Linear and Nonlinear Waves (1974), Springer: Springer New York · Zbl 0373.76001
[116] Wazwaz, A. M., Partial Differential Equations: Methods and Applications (2002), Balkema Publishers: Balkema Publishers The Netherlands · Zbl 1079.35001
[117] Wazwaz, A. M., A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simul., 56, 269-276 (2001) · Zbl 0999.65109
[118] Wazwaz, A. M., The modified decomposition method and Pade approximations for solving the Thomas-Fermi equation, Appl. Math. Comput., 105, 11-29 (1999) · Zbl 0956.65064
[119] Wazwaz, A. M., Analytical study on Burgers, Fisher, Huxley equations and combined forms of these equations, Appl. Math. Comput., 195, 754-761 (2008) · Zbl 1132.65098
[120] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[121] Bekir, A., New solitons and periodic wave solutions for some nonlinear physical models by using sine-cosine method, Phys. Scr., 77, 4, 501 (2008) · Zbl 1306.76009
[122] Bekir, A.; Boz, A., Exact solutions for nonlinear evolution equations using Expfunction method, Phys. Lett. A, 372, 10, 1619 (2008) · Zbl 1217.35151
[123] Dhawan, S.; Kapoor, S.; Kumar, S.; Rawat, S., Contemporary review of techniques for the solution of nonlinear Burgers equation, Journal of Computer Science, 3, 5, 405-419 (2012)
[124] Feng, Z.; Li, Yang, Complex traveling wave solutions to the Fisher equation, Phys. A, 366, 115-123 (2006)
[125] Gorguis, A., A comparison between Cole-Hopf transformation and the decomposition method for solving Burgers’ equations, Appl. Math. Comput. (2008)
[126] Gourley, S. A., Travelling front solutions of a non-local Fisher equation, J. Math. Biol., 41, 272-284 (2000) · Zbl 0982.92028
[127] Hirota, R., Direct method of finding exact solutions of nonlinear evolution equations, (Bullough, R.; Caudrey, P., Bäcklund Transformations (1980), Springer: Springer Berlin), 1157
[128] Kawahara, T.; Tanaka, M., Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation, Phys. Lett. A, 97, 8, 311-314 (1983)
[129] Geng, Wanhai.; Chen, Yiming.; Li, Yulian.; Wang, Dong., Wavelet Method for Nonlinear Partial Differential Equations of Fractional Order, Computer and Information Science, 4, 5, 28-35 (2011)
[130] Wang, X. Y., Exact and explicit solitary wave solutions for the generalized Fisher’s equation, Phys. Lett. A, 131, 4/5, 277-279 (1988)
[131] Zhi, S., Cao Yong-yan, Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method, Appl. Math. Model., 36, 11, 5143-5161 (2012) · Zbl 1254.65138
[132] Mahalakshmi, M.; Rajaraman, R.; Hariharan, G.; Kannan, K., Approximate analytical solutions of two dimensional transient heat conduction problems, Appl. Math. Sci., 71, 3507-3518 (2012) · Zbl 1264.35015
[133] Barinka, A.; Barsch, T.; Charton, P.; Cohen, A.; Dahlke, S.; Dahmen, W.; Urban, K., Adaptive wavelet schemes for elliptic problems - implementation and numerical experiments, SIAM J. Sci. Comput., 23, 910 (2001) · Zbl 1016.65090
[134] Cattani, C., Wave propagation of Shannon wavelets, ICISA, 7781-7799 (2006)
[135] Hashim, I.; Noorani, M. S.M.; Said Al-Hadidi, M. R., Solving the generalized Burgers-Huxley equation using the Adomian decomposition method, Math. Comput. Model., 43, 11-12, 1404-1411 (2006) · Zbl 1133.65083
[136] Farge, M.; Hunt, J. C.R.; Vassilicos, J. C., Wavelets, Fractals and Fourier Transforms (1993), Clarendon: Clarendon New York · Zbl 0978.42504
[140] Hariharan, G.; Kannan, K., An Overview of Haar Wavelet Method for Solving Differential and Integral Equations, World Applied Sciences Journal, 23, 12, 1-14 (2013)
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.