# Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Results

## 3. The Properties of the Shape Functions and Solutions

## 4. The Diffusion Equation with Constant Source

## 5. Summary and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## References

- Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, UK, 1956. [Google Scholar]
- Ghez, R. Diffusion Phenomena; Dover Publication: Mineola, NY, USA, 2001. [Google Scholar]
- Bennett, T.D. Transport by Advection and Diffusion: Momentum, Heat and Mass Transfer; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Lienhard, J.H., IV; Lienhard, J.H., V. A Heat Transfer Textbook, 4th ed.; Phlogiston Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Newman, J.; Battaglia, V. The Newman Lectures on Transport Phenomena; Jenny Stanford Publishing: Dubai, United Arab Emirates, 2021. [Google Scholar]
- Kardar, M.; Parisi, G.; Zhang, Y.-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett.
**1986**, 56, 889. [Google Scholar] [CrossRef] - Barna, I.F.; Bognár, G.; Guedda, M.; Hriczó, K.; Mátyás, L. Analytic self-similar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise terms. Math. Model. Anal.
**2020**, 25, 241. [Google Scholar] [CrossRef] - Barabási, A.L.; Stanley, E. Fractal Concepts in Surface Growth; Cambridge University Press: Cambridge, MA, USA, 1995. [Google Scholar]
- Mátyás, L.; Gaspard, P. Entropy production in diffusion-reaction systems: The reactive random Lorentz gas. Phys. Rev. E
**2005**, 71, 036147. [Google Scholar] [CrossRef] [PubMed] - Mátyás, L.; Barna, I.F. General self-similar solutions of diffusion equation and related constructions. Rom. J. Phys.
**2022**, 67, 101. [Google Scholar] - Barna, I.F.; Mátyás, L. Advanced Analytic Self-Similar Solutions of Regular and Irregular Diffusion Equations. Mathematics
**2022**, 10, 3281. [Google Scholar] [CrossRef] - Cannon, J.R. The One-Dimensional Heat Equation; Addison-Wesley Publishing: Reading, MA, USA, 1984. [Google Scholar]
- Cole, K.D.; Beck, J.V.; Haji-Sheikh, A.; Litkouhi, B. Heat Conduction Using Green’s Functions; Series in Computational and Physical Processes in Mechanics and Thermal Sciences (2nd ed); CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Mátyás, L.; Tél, T.; Vollmer, J. Thermodynamic cross effects from dynamical systems. Phys. Rev. E
**2000**, 61, R3295. [Google Scholar] [CrossRef] - Thambynayagam, R.K.M. The Diffusion Handbook: Applied Solutions for Engineers; McGraw-Hill: New York, NY, USA, 2011. [Google Scholar]
- Michaud, G.; Alecian, G.; Richer, G. Atomic Diffusion in Stars; Springer: New York, NY, USA, 2013; Volume 70. [Google Scholar]
- Murray, J.D. Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd ed.; Springer: New York, NY, USA, 2003. [Google Scholar]
- Alebraheem, J. Predator interference in a predator-prey model with mixed functional and numerical responses. Hindawi J. Math.
**2023**, 2023, 4349573. [Google Scholar] [CrossRef] - Perthame, B. Parabolic Equations in Biology; Springer International Publishing: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Szép, R.; Mateescu, E.; Nechifor, A.C.; Keresztesi, Á. Chemical characteristics and source analysis on ionic composition of rainwater collected in the Charpatians “Cold Pole”, Ciuc basin, Eastern Carpatians, Romania. Environ. Sci. Pollut. Res.
**2017**, 24, 27288. [Google Scholar] [CrossRef] [PubMed] - Gillespie, D.T.; Seitaridou, E. Simple Brownian Diffusion; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Tálos, K.; Páger, C.; Tonk, S.; Majdik, C.; Kocsis, B.; Kilár, F.; Pernyeszi, T. Cadmium biosorption on native Saccharomyces cerevisiae cells in aqueous suspension. Acta Univ. Sapientiae Agric. Environ.
**2009**, 1, 20. [Google Scholar] - Nechifor, G.; Voicu, S.I.; Nechifor, A.C.; Garea, S. Nanostructured hybrid membrane polysulfone-carbon nanotubes for hemodialysis. Desalination
**2009**, 241, 342. [Google Scholar] [CrossRef] - Lv, J.; Ren, K.; Chen, Y. CO
_{2}diffusion in various carbonated beverages: A molecular dynamic study. Phys. Chem.**2018**, 122, 1655. [Google Scholar] [CrossRef] - Hägerstrand, T. Innovation Diffusion as a Spatial Process; The University of Chicago Press: Chicago, IL, USA, 1967. [Google Scholar]
- Rogers, E.M. Diffusion of Innovations; The Free Press: Los Angeles, CA, USA, 1983. [Google Scholar]
- Nakicenovic, N.; Griübler, A. Diffusion of Technologies and Social Behavior; Springer: Berlin/Heidelberg, Germany, 1991. [Google Scholar]
- Bunde, A.; Kärger, J.C.; Vogl, G. Diffusive Spreading in Nature, Technology and Society; Springer: Berlin/Heidelberg, Germany, 2018. [Google Scholar]
- Vogel, G. Adventure Diffusion; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Mazzoni, T. A First Course in Quantitative Finance; Cambridge University Press: Cambridge, MA, USA, 2018. [Google Scholar]
- Lázár, E. Quantifying the economic value of warranties: A survey. Acta Univ. Sapientiae Econ. Bus.
**2014**, 2, 75. [Google Scholar] [CrossRef] - Albert, R.; Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys.
**2002**, 74, 47. [Google Scholar] [CrossRef] - Rogolino, P.; Kovács, R.; Ván, P.; Cimmelli, V.A. Generalized heat-transport equations: Parabolic and hyperbolic models. Contin. Mech. Thermodyn.
**2018**, 30, 1245. [Google Scholar] [CrossRef] - Jalghaf, H.K.; Kovács, E.; Majár, J.; Nagy, A.; Askar, A.H. Explicit stable finite difference methods for diffusion-reaction type equations. Mathematics
**2021**, 9, 3308. [Google Scholar] [CrossRef] - Nagy, A.; Saleh, M.; Omle, I.; Kareem, H.; Kovács, E. New stable, explicit shifted-hopscotch algoritms for the heat equation. Math. Comput. Appl.
**2021**, 26, 61. [Google Scholar] - Ezzahri, Y.; Ordonez-Miranda, J.; Joulain, K. Heat transport in semiconductor crystals under large temperature gradients. Int. J. Heat Mass Transf.
**2017**, 108, 1357. [Google Scholar] [CrossRef] - Cussler, E.L. Diffusion: Mass Transfer in Fluid Systems, 3rd ed.; Cambridge University Press: Cambridge, MA, USA, 2009. [Google Scholar]
- Bluman, G.W.; Cole, J.D. The general similarity solution of the heat equation. J. Math. Mech.
**1969**, 18, 1025. [Google Scholar] - Sedov, L. Similarity and Dimensional Methods in Mechanics; CRC Press: Boca Raton, FL, USA, 1993. [Google Scholar]
- Zel’dovich, Y.B.; Raizer, Y.P. Physics of Shock Waves and High Temperature Hydrodynamic Phenomena; Academic Press: New York, NY, USA, 1966. [Google Scholar]
- Baraneblatt, G.I. Similarity, Self-Similarity, and Intermediate Asymptotics; Consultants Bureau: New York, NY, USA, 1979. [Google Scholar]
- Barna, I.F.; Mátyás, L. Analytic solutions for the three dimensional compressible Navier-Stokes equation. Fluid Dyn. Res.
**2014**, 46, 055508. [Google Scholar] [CrossRef] - Barna, I.F.; Pocsai, M.A.; Lökös, S.; Mátyás, L. Rayleigh-Benard convection in the generalized Oberbeck-Boussinesq system. Chaos Solitons Fractals
**2017**, 103, 336. [Google Scholar] [CrossRef] - Barna, I.F.; Pocsai, M.A.; Barnaföldi, G.G. Self-similar solutions of a gravitating dark fluid. Mathematics
**2022**, 10, 3220. [Google Scholar] [CrossRef] - Barna, I.F.; Pocsai, M.A.; Mátyás, L. Analytic solutions of the Madelung equation. J. Gen. Lie Theory Appl.
**2017**, 11, 1000271. [Google Scholar] - Csanád, M.; Vargyas, M. Observables from a solution of (1+3)-dimensional relativistic hydrodynamics. Eur. Phys. J. A
**2010**, 44, 473. [Google Scholar] [CrossRef] - Barna, I.F.; Mátyás, L. Analytic solutions for the one-dimensional compressible Euler equation with heat conduction closed with different kind of equation of states. Miskolc Math. Notes
**2013**, 14, 785. [Google Scholar] [CrossRef] - Nath, G.; Singh, S. Approximate analytical solution for shock wave in rotational axisymetric perfect gas with azimutal magnetic field: Isotermal flow. J. Astrophys. Astron.
**2019**, 40, 50. [Google Scholar] [CrossRef] - Sahu, P.K. Shock wave driven out by a piston in a mixture of a non-ideal gas and small solid particles under the influence of azimuthal or axial magnetic field. Braz. J. Phys.
**2020**, 50, 548. [Google Scholar] [CrossRef] - Kanchana, C.; Su, Y.; Zhao, Y. Primary and secondary instabilities in Rayleigh-Benard convention of water-copper nanoliquid. Commun. Nonlinear Sci. Numer. Simul.
**2020**, 83, 105129. [Google Scholar] [CrossRef] - Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. NIST Handbook of Mathematical Functions; Cambridge University Press: Cambridge, MA, USA, 2010. [Google Scholar]
- Kythe, P.K. Green’s Functions and Linear Differential Equations; Chapman & Hall/CRC Applied Mathematics and Nonliner Science; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Rother, T. Green’s Functions in Classical Physics; Lecture Notes in Physics; Springer International Publishing: New York, NY, USA, 2017; Volume 938. [Google Scholar]
- Bronshtein, I.N.; Semendyayev, K.A.; Musiol, G.; Mühlig, H. Handbook of Mathematics; Springer: Wiesbaden, Germany, 2007. [Google Scholar]
- Greiner, W.; Reinhardt, J. Quantum Electrodynamics; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Claus, I.; Gaspard, P. Fractals and dynamical chaos in a two-dimensional Lorentz gas with sinks. Phys. Rev. E
**2001**, 63, 036227. [Google Scholar] [CrossRef] - Rápó, E.; Tonk, S. Factors affecting synthetic dye adsorption; desorption studies: A review of results from the last five years (2017–2021). Molecules
**2021**, 26, 5419. [Google Scholar] [CrossRef] - Boltzmann, L. Zur Intergration der Diffusionsgleichung bei variabeln Diffusionscoefficienten. Ann. Phys.
**1894**, 53, 959. [Google Scholar] [CrossRef] - Lonngren, K.E. Self similar solution of plasma equations. Proc. Indian Acad. Sci.
**1977**, 86, 125. [Google Scholar] [CrossRef] - Mehrer, H.; Stolwijk, N.A. Heroes and highlights in the history of diffusion. Diffus. Fundam.
**2009**, 11, 1. [Google Scholar]

**Figure 1.**The importance of $\alpha $ and $\beta $ in case of the change of variables of Equation (5).

**Figure 2.**The solution $C(x,t)$ for (

**a**) $\alpha =1/2$, (

**b**) $\alpha =3/2$ (

**c**) $\alpha =5/2$ and (

**d**) $\alpha =7/2$, respectively.

**Figure 3.**Even shape functions $f\left(\eta \right)$ of Equation (16) for three different self-similar $\alpha $ exponents. The black, blue and red curves are for $\alpha =1/2,3/2$ and $5/2$ numerical values, with the same diffusion constant (D = 2), respectively. Note that shape functions with larger $\alpha $s have more zero transitions. We will show that for $\alpha >0$ integer values, the integral of the shape functions give zero on the whole and the half-axis as well.

**Figure 7.**The shape function $C(x,t)$, solution of Equation (63), for $D=1$ and $n=1$, in case (

**a**) ${\kappa}_{1}=0.1$${\kappa}_{2}=0.03$ and (

**b**) ${\kappa}_{1}=0.2$${\kappa}_{2}=0.2$.

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**MDPI and ACS Style**

Mátyás, L.; Barna, I.F.
Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon. *Universe* **2023**, *9*, 264.
https://doi.org/10.3390/universe9060264

**AMA Style**

Mátyás L, Barna IF.
Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon. *Universe*. 2023; 9(6):264.
https://doi.org/10.3390/universe9060264

**Chicago/Turabian Style**

Mátyás, László, and Imre Ferenc Barna.
2023. "Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon" *Universe* 9, no. 6: 264.
https://doi.org/10.3390/universe9060264