# Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

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## Abstract

**:**

## 1. Introduction

## 2. Analytical and Numerical Matching for the (2+1) D-Fisher-KPP Model

#### 2.1. MDA Analytical Versus TQBS Numerical Techniques along (2+1) D-Fisher-KPP Model

**Set I**

**Set II**

**Set III**

#### Matching between Analytical and Numerical

#### 2.2. Kud Analytical vs. TQBS Numerical Techniques along (2+1) D-Fisher-KPP Model

**Set I**

**Set II**

**Set III**

#### Semi-Analytical Solutions

## 3. Results’ Interpretation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Helliar, C.V.; Crawford, L.; Rocca, L.; Teodori, C.; Veneziani, M. Permissionless and permissioned blockchain diffusion. Int. J. Inf. Manag.
**2020**, 54, 102136. [Google Scholar] [CrossRef] - Axpe, E.; Chan, D.; Offeddu, G.S.; Chang, Y.; Merida, D.; Hernandez, H.L.; Appel, E.A. A multiscale model for solute diffusion in hydrogels. Macromolecules
**2019**, 52, 6889–6897. [Google Scholar] [CrossRef] [Green Version] - Chandrabose, S.; Chen, K.; Barker, A.J.; Sutton, J.J.; Prasad, S.K.; Zhu, J.; Zhou, J.; Gordon, K.C.; Xie, Z.; Zhan, X.; et al. High exciton diffusion coefficients in fused ring electron acceptor films. J. Am. Chem. Soc.
**2019**, 141, 6922–6929. [Google Scholar] [CrossRef] - Grebenkov, D.S. Paradigm shift in diffusion-mediated surface phenomena. Phys. Rev. Lett.
**2020**, 125, 078102. [Google Scholar] [CrossRef] [PubMed] - Iima, M.; Honda, M.; Sigmund, E.E.; Ohno Kishimoto, A.; Kataoka, M.; Togashi, K. Diffusion MRI of the breast: Current status and future directions. J. Magn. Reson. Imaging
**2020**, 52, 70–90. [Google Scholar] [CrossRef] [PubMed] - Brown, J.L.; Johnston, W.; Delaney, C.; Rajendran, R.; Butcher, J.; Khan, S.; Bradshaw, D.; Ramage, G.; Culshaw, S. Biofilm-stimulated epithelium modulates the inflammatory responses in co-cultured immune cells. Sci. Rep.
**2019**, 9, 1–14. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hosseininia, M.; Heydari, M. Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction–diffusion equation involving Mittag–Leffler non-singular kernel. Chaos Solitons Fractals
**2019**, 127, 400–407. [Google Scholar] [CrossRef] - García-Crespo, C.; Soria, M.E.; Gallego, I.; Ávila, A.I.d.; Martínez-González, B.; Vázquez-Sirvent, L.; Gómez, J.; Briones, C.; Gregori, J.; Quer, J.; et al. Dissimilar conservation pattern in hepatitis C virus mutant spectra, consensus sequences, and data banks. J. Clin. Med.
**2020**, 9, 3450. [Google Scholar] [CrossRef] - Yang, X.; Li, Y.; Zou, L.; Zhu, Z. Role of exosomes in crosstalk between cancer-associated fibroblasts and cancer cells. Front. Oncol.
**2019**, 9, 356. [Google Scholar] [CrossRef] [PubMed] - Bochner, S. Diffusion equation and stochastic processes. Proc. Natl. Acad. Sci. USA
**1949**, 35, 368. [Google Scholar] [CrossRef] [Green Version] - Elphick, C.; Coullet, P.; Repaux, D. Nature of spatial chaos. Phys. Rev. Lett
**1987**, 58, 431–434. [Google Scholar] - Dee, G.; van Saarloos, W. Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett.
**1988**, 60, 2641. [Google Scholar] [CrossRef] [Green Version] - Van Saarloos, W. Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence. Phys. Rev. A
**1989**, 39, 6367. [Google Scholar] [CrossRef] - El-Hachem, M.; McCue, S.W.; Jin, W.; Du, Y.; Simpson, M.J. Revisiting the Fisher–Kolmogorov–Petrovsky–Piskunov equation to interpret the spreading–extinction dichotomy. Proc. R. Soc. A
**2019**, 475, 20190378. [Google Scholar] [CrossRef] [Green Version] - Fisher, R.A. The wave of advance of advantageous genes. Ann. Eugen.
**1937**, 7, 355–369. [Google Scholar] [CrossRef] [Green Version] - Kolmogorov, A.N.; Petrovskii, I.G.; Piskunov, N.S. A study of the diffusion equation with increase in the amount of substance and its application to a biology problem. Bull. Univ. Mosc. Ser. Int.
**1937**, 1, 1–16. [Google Scholar] - Veeresha, P.; Prakasha, D.; Singh, J.; Khan, I.; Kumar, D. Analytical approach for fractional extended Fisher–Kolmogorov equation with Mittag-Leffler kernel. Adv. Differ. Equ.
**2020**, 2020, 1–17. [Google Scholar] [CrossRef] - Levchenko, E.A.; Shapovalov, A.V.; Trifonov, A.Y. Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation. J. Math. Anal. Appl.
**2012**, 395, 716–726. [Google Scholar] [CrossRef] [Green Version] - Luther, R. II. Sitzung am Dienstag, den 22. Mai, vormittags 9 Uhr, im grossen Auditorium des chemischen Laboratoriums der Technischen Hochschule. Räumliche Fortpflanzung chemischer Reaktionen. Z. Elektrochem. Angew. Phys. Chem.
**1906**, 12, 596–600. [Google Scholar] [CrossRef] - Oruç, Ö. An efficient wavelet collocation method for nonlinear two-space dimensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation. Eng. Comput.
**2020**, 36, 839–856. [Google Scholar] [CrossRef] - Morgado, G.; Nowakowski, B.; Lemarchand, A. Fisher-Kolmogorov-Petrovskii-Piskunov wave front as a sensor of perturbed diffusion in concentrated systems. Phys. Rev. E
**2019**, 99, 022205. [Google Scholar] [CrossRef] [Green Version] - El-Nabulsi, R.A. Fourth-Order Ginzburg-Landau differential equation a la Fisher-Kolmogorov and quantum aspects of superconductivity. Phys. C Supercond. Its Appl.
**2019**, 567, 1353545. [Google Scholar] [CrossRef] - El-Nabulsi, R.A. Orbital dynamics satisfying the 4 th-order stationary extended Fisher-Kolmogorov equation. Astrodynamics
**2020**, 4, 31–39. [Google Scholar] [CrossRef] - Khater, M.M.; Seadawy, A.R.; Lu, D. Bifurcations of solitary wave solutions for (two and three)-dimensional nonlinear partial differential equation in quantum and magnetized plasma by using two different methods. Results Phys.
**2018**, 9, 142–150. [Google Scholar] [CrossRef] - Khater, M.; Attia, R.A.; Abdel-Aty, A.H.; Abdel-Khalek, S.; Al-Hadeethi, Y.; Lu, D. On the computational and numerical solutions of the transmission of nerve impulses of an excitable system (the neuron system). J. Intell. Fuzzy Syst.
**2020**, 38, 2603–2610. [Google Scholar] [CrossRef] - Qin, H.; Attia, R.A.; Khater, M.; Lu, D. Ample soliton waves for the crystal lattice formation of the conformable time-fractional (N+ 1) Sinh-Gordon equation by the modified Khater method and the Painlevé property. J. Intell. Fuzzy Syst.
**2020**, 38, 2745–2752. [Google Scholar] [CrossRef] - Khater, M.M.; Baleanu, D. On abundant new solutions of two fractional complex models. Adv. Differ. Equ.
**2020**, 2020, 1–14. [Google Scholar] [CrossRef] - Yue, C.; Khater, M.M.; Attia, R.A.; Lu, D. The plethora of explicit solutions of the fractional KS equation through liquid–gas bubbles mix under the thermodynamic conditions via Atangana–Baleanu derivative operator. Adv. Differ. Equ.
**2020**, 2020, 1–12. [Google Scholar] [CrossRef] [Green Version] - Abdel-Aty, A.H.; Khater, M.M.; Dutta, H.; Bouslimi, J.; Omri, M. Computational solutions of the HIV-1 infection of CD4+ T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy. Chaos Solitons Fractals
**2020**, 139, 110092. [Google Scholar] [CrossRef] [PubMed] - Khater, M.M.; Attia, R.A.; Park, C.; Lu, D. On the numerical investigation of the interaction in plasma between (high & low) frequency of (Langmuir & ion-acoustic) waves. Results Phys.
**2020**, 18, 103317. [Google Scholar] - Khater, M.M.; Attia, R.A.; Abdel-Aty, A.H.; Alharbi, W.; Lu, D. Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms. Chaos Solitons Fractals
**2020**, 136, 109824. [Google Scholar] [CrossRef] - Abdel-Aty, A.H.; Khater, M.M.; Attia, R.A.; Abdel-Aty, M.; Eleuch, H. On the new explicit solutions of the fractional nonlinear space-time nuclear model. Fractals
**2020**, 28, 2040035. [Google Scholar] [CrossRef] - Qin, H.; Khater, M.; Attia, R.A. Inelastic Interaction and Blowup New Solutions of Nonlinear and Dispersive Long Gravity Waves. J. Funct. Spaces
**2020**, 2020, 5362989. [Google Scholar] [CrossRef] - Park, C.; Khater, M.M.; Abdel-Aty, A.H.; Attia, R.A.; Rezazadeh, H.; Zidan, A.; Mohamed, A.B. Dynamical analysis of the nonlinear complex fractional emerging telecommunication model with higher–order dispersive cubic–quintic. Alex. Eng. J.
**2020**, 59, 1425–1433. [Google Scholar] [CrossRef] - Khater, M.M.; Ghanbari, B.; Nisar, K.S.; Kumar, D. Novel exact solutions of the fractional Bogoyavlensky–Konopelchenko equation involving the Atangana-Baleanu-Riemann derivative. Alex. Eng. J.
**2020**, 59, 2957–2967. [Google Scholar] [CrossRef] - Yue, C.; Lu, D.; Khater, M.M.; Abdel-Aty, A.H.; Alharbi, W.; Attia, R.A. On explicit wave solutions of the fractional nonlinear DSW system via the modified Khater method. Fractals
**2020**, 28, 2040034. [Google Scholar] [CrossRef] - Abdel-Aty, A.H.; Khater, M.; Attia, R.A.; Eleuch, H. Exact Traveling and Nano-Solitons Wave Solitons of the Ionic Waves Propagating along Microtubules in Living Cells. Mathematics
**2020**, 8, 697. [Google Scholar] [CrossRef] - Qin, H.; Khater, M.; Attia, R.A. Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids. Math. Probl. Eng.
**2020**, 2020, 3498796. [Google Scholar] [CrossRef] - Demiray, S.T.; Bayrakci, U. Soliton solutions for space-time fractional Heisenberg ferromagnetic spin chain equation by generalized Kudryashov method and modified exp-expansion function method. Rev. Mex. Física
**2021**, 67, 393–402. [Google Scholar] [CrossRef] - Chu, Y.; Khater, M.M.; Hamed, Y. Diverse novel analytical and semi-analytical wave solutions of the generalized (2+1)-dimensional shallow water waves model. AIP Adv.
**2021**, 11, 015223. [Google Scholar] [CrossRef] - Shakeel, M. Travelling wave solution of the Fisher-Kolmogorov equation with non-linear diffusion. Appl. Math.
**2013**, 4, 35319. [Google Scholar] [CrossRef]

**Figure 7.**Two-dimensional (

**a**) and radar (

**b**) plots for the calculated absolute error along the MDA, MKud. analytical schemes and the TQBS numerical scheme.

Value of x | Analytical | Numerical | Error | Value of x | Analytical | Numerical | Error |
---|---|---|---|---|---|---|---|

0 | 0.5 | 0.5 | 0 | 0.515625 | 0.925947 | 0.999254 | 0.073307 |

0.015625 | 0.519127 | 0.596515 | 0.077388 | 0.53125 | 0.931028 | 0.999203 | 0.068176 |

0.03125 | 0.538199 | 0.705602 | 0.167403 | 0.546875 | 0.935784 | 0.999146 | 0.063362 |

0.046875 | 0.557159 | 0.77828 | 0.221121 | 0.5625 | 0.940233 | 0.999082 | 0.058849 |

0.0625 | 0.575954 | 0.835791 | 0.259837 | 0.578125 | 0.944392 | 0.999013 | 0.05462 |

0.078125 | 0.594532 | 0.878033 | 0.283501 | 0.59375 | 0.948278 | 0.998936 | 0.050658 |

0.09375 | 0.612843 | 0.909745 | 0.296901 | 0.609375 | 0.951906 | 0.998854 | 0.046947 |

0.109375 | 0.630841 | 0.933241 | 0.3024 | 0.625 | 0.955292 | 0.998764 | 0.043472 |

0.125 | 0.648482 | 0.950674 | 0.302192 | 0.640625 | 0.95845 | 0.998667 | 0.040217 |

0.140625 | 0.665726 | 0.963567 | 0.297841 | 0.65625 | 0.961393 | 0.998561 | 0.037168 |

0.15625 | 0.682539 | 0.973094 | 0.290555 | 0.671875 | 0.964136 | 0.998448 | 0.034312 |

0.171875 | 0.698889 | 0.980124 | 0.281235 | 0.6875 | 0.966691 | 0.998325 | 0.031634 |

0.1875 | 0.714748 | 0.985307 | 0.270559 | 0.703125 | 0.96907 | 0.998192 | 0.029123 |

0.203125 | 0.730095 | 0.989125 | 0.25903 | 0.71875 | 0.971283 | 0.998049 | 0.026765 |

0.21875 | 0.744911 | 0.991936 | 0.247025 | 0.734375 | 0.973343 | 0.997894 | 0.024551 |

0.234375 | 0.759182 | 0.994003 | 0.234821 | 0.75 | 0.975259 | 0.997727 | 0.022468 |

0.25 | 0.772897 | 0.995521 | 0.222623 | 0.765625 | 0.97704 | 0.997547 | 0.020506 |

0.265625 | 0.786052 | 0.996634 | 0.210582 | 0.78125 | 0.978696 | 0.997352 | 0.018656 |

0.28125 | 0.798644 | 0.997448 | 0.198805 | 0.796875 | 0.980235 | 0.997142 | 0.016907 |

0.296875 | 0.810672 | 0.998042 | 0.18737 | 0.8125 | 0.981665 | 0.996915 | 0.015251 |

0.3125 | 0.822143 | 0.998474 | 0.176331 | 0.828125 | 0.982993 | 0.996671 | 0.013678 |

0.328125 | 0.833061 | 0.998784 | 0.165724 | 0.84375 | 0.984226 | 0.996407 | 0.012181 |

0.34375 | 0.843437 | 0.999006 | 0.155569 | 0.859375 | 0.985372 | 0.996122 | 0.010751 |

0.359375 | 0.853281 | 0.999161 | 0.14588 | 0.875 | 0.986435 | 0.995815 | 0.00938 |

0.375 | 0.862607 | 0.999267 | 0.13666 | 0.890625 | 0.987422 | 0.995484 | 0.008062 |

0.390625 | 0.87143 | 0.999336 | 0.127906 | 0.90625 | 0.988338 | 0.995126 | 0.006788 |

0.40625 | 0.879765 | 0.999377 | 0.119611 | 0.921875 | 0.989188 | 0.99474 | 0.005552 |

0.421875 | 0.88763 | 0.999396 | 0.111766 | 0.9375 | 0.989977 | 0.994326 | 0.004349 |

0.4375 | 0.895041 | 0.999399 | 0.104357 | 0.953125 | 0.990709 | 0.993871 | 0.003162 |

0.453125 | 0.902018 | 0.999388 | 0.09737 | 0.96875 | 0.991387 | 0.993411 | 0.002023 |

0.46875 | 0.908578 | 0.999367 | 0.090789 | 0.984375 | 0.992017 | 0.992795 | 0.000778 |

0.484375 | 0.914741 | 0.999337 | 0.084596 | 1 | 0.992601 | 0.992601 | $5.55\times {10}^{-17}$ |

Value of x | Analytical | Numerical | Error | Value of x | Analytical | Numerical | Error |
---|---|---|---|---|---|---|---|

0 | 0.5 | 0.5 | 0 | 0.515625 | 0.373876 | 0.373876 | 1.62 $\times {10}^{-11}$ |

0.015625 | 0.496094 | 0.496094 | 1.2 $\times {10}^{-12}$ | 0.53125 | 0.370225 | 0.370225 | 1.59 $\times {10}^{-11}$ |

0.03125 | 0.492188 | 0.492188 | 2.94 $\times {10}^{-12}$ | 0.546875 | 0.36659 | 0.36659 | 1.56 $\times {10}^{-11}$ |

0.046875 | 0.488283 | 0.488283 | 4.39 $\times {10}^{-12}$ | 0.5625 | 0.362969 | 0.362969 | 1.52 $\times {10}^{-11}$ |

0.0625 | 0.48438 | 0.48438 | 5.79 $\times {10}^{-12}$ | 0.578125 | 0.359364 | 0.359364 | 1.49 $\times {10}^{-11}$ |

0.078125 | 0.480479 | 0.480479 | 7.07 $\times {10}^{-12}$ | 0.59375 | 0.355775 | 0.355775 | 1.45 $\times {10}^{-11}$ |

0.09375 | 0.47658 | 0.47658 | 8.26 $\times {10}^{-12}$ | 0.609375 | 0.352202 | 0.352202 | 1.41 $\times {10}^{-11}$ |

0.109375 | 0.472683 | 0.472683 | 9.37 $\times {10}^{-12}$ | 0.625 | 0.348645 | 0.348645 | 1.37 $\times {10}^{-11}$ |

0.125 | 0.468791 | 0.468791 | 1.04 $\times {10}^{-11}$ | 0.640625 | 0.345105 | 0.345105 | 1.32 $\times {10}^{-11}$ |

0.140625 | 0.464902 | 0.464902 | 1.13 $\times {10}^{-11}$ | 0.65625 | 0.341582 | 0.341582 | 1.28 $\times {10}^{-11}$ |

0.15625 | 0.461017 | 0.461017 | 1.22 $\times {10}^{-11}$ | 0.671875 | 0.338077 | 0.338077 | 1.23 $\times {10}^{-11}$ |

0.171875 | 0.457137 | 0.457137 | 1.29 $\times {10}^{-11}$ | 0.6875 | 0.334589 | 0.334589 | 1.18 $\times {10}^{-11}$ |

0.1875 | 0.453262 | 0.453262 | 1.36 $\times {10}^{-11}$ | 0.703125 | 0.33112 | 0.33112 | 1.13 $\times {10}^{-11}$ |

0.203125 | 0.449393 | 0.449393 | 1.43 $\times {10}^{-11}$ | 0.71875 | 0.327668 | 0.327668 | 1.08 $\times {10}^{-11}$ |

0.21875 | 0.44553 | 0.44553 | 1.48 $\times {10}^{-11}$ | 0.734375 | 0.324235 | 0.324235 | 1.02 $\times {10}^{-11}$ |

0.234375 | 0.441673 | 0.441673 | 1.53 $\times {10}^{-11}$ | 0.75 | 0.320821 | 0.320821 | 9.68 $\times {10}^{-12}$ |

0.25 | 0.437823 | 0.437823 | 1.58 $\times {10}^{-11}$ | 0.765625 | 0.317426 | 0.317426 | 9.13 $\times {10}^{-12}$ |

0.265625 | 0.433981 | 0.433981 | 1.62 $\times {10}^{-11}$ | 0.78125 | 0.314051 | 0.314051 | 8.56 $\times {10}^{-12}$ |

0.28125 | 0.430147 | 0.430147 | 1.65 $\times {10}^{-11}$ | 0.796875 | 0.310694 | 0.310694 | 7.98 $\times {10}^{-12}$ |

0.296875 | 0.426322 | 0.426322 | 1.68 $\times {10}^{-11}$ | 0.8125 | 0.307358 | 0.307358 | 7.39 $\times {10}^{-12}$ |

0.3125 | 0.422505 | 0.422505 | 1.7 $\times {10}^{-11}$ | 0.828125 | 0.304042 | 0.304042 | 6.8 $\times {10}^{-12}$ |

0.328125 | 0.418697 | 0.418697 | 1.72 $\times {10}^{-11}$ | 0.84375 | 0.300746 | 0.300746 | 6.19 $\times {10}^{-12}$ |

0.34375 | 0.414899 | 0.414899 | 1.73 $\times {10}^{-11}$ | 0.859375 | 0.29747 | 0.29747 | 5.58 $\times {10}^{-12}$ |

0.359375 | 0.411111 | 0.411111 | 1.74 $\times {10}^{-11}$ | 0.875 | 0.294215 | 0.294215 | 4.96 $\times {10}^{-12}$ |

0.375 | 0.407333 | 0.407333 | 1.74 $\times {10}^{-11}$ | 0.890625 | 0.290981 | 0.290981 | 4.34 $\times {10}^{-12}$ |

0.390625 | 0.403567 | 0.403567 | 1.74 $\times {10}^{-11}$ | 0.90625 | 0.287768 | 0.287768 | 3.71 $\times {10}^{-12}$ |

0.40625 | 0.399812 | 0.399812 | 1.74 $\times {10}^{-11}$ | 0.921875 | 0.284576 | 0.284576 | 3.07 $\times {10}^{-12}$ |

0.421875 | 0.396068 | 0.396068 | 1.73 $\times {10}^{-11}$ | 0.9375 | 0.281406 | 0.281406 | 2.44 $\times {10}^{-12}$ |

0.4375 | 0.392337 | 0.392337 | 1.72 $\times {10}^{-11}$ | 0.953125 | 0.278257 | 0.278257 | 1.79 $\times {10}^{-12}$ |

0.453125 | 0.388618 | 0.388618 | 1.71 $\times {10}^{-11}$ | 0.96875 | 0.27513 | 0.27513 | 1.16 $\times {10}^{-12}$ |

0.46875 | 0.384912 | 0.384912 | 1.69 $\times {10}^{-11}$ | 0.984375 | 0.272025 | 0.272025 | 4.57 $\times {10}^{-13}$ |

0.484375 | 0.38122 | 0.38122 | 1.67 $\times {10}^{-11}$ | 1 | 0.268941 | 0.268941 | 5.55 $\times {10}^{-17}$ |

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**MDPI and ACS Style**

Khater, M.M.A.; Alabdali, A.M.
Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation. *Mathematics* **2021**, *9*, 1440.
https://doi.org/10.3390/math9121440

**AMA Style**

Khater MMA, Alabdali AM.
Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation. *Mathematics*. 2021; 9(12):1440.
https://doi.org/10.3390/math9121440

**Chicago/Turabian Style**

Khater, Mostafa M. A., and Aliaa Mahfooz Alabdali.
2021. "Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation" *Mathematics* 9, no. 12: 1440.
https://doi.org/10.3390/math9121440