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Functional separable solutions of nonlinear reaction–diffusion equations with variable coefficients

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  • Polyanin, Andrei D.

Abstract

The paper presents a number of new functional separable solutions to nonlinear reaction–diffusion equations of the formc(x)ut=[a(x)ux]x+b(x)ux+p(x)f(u),where f(u) is an arbitrary function. It is shown that any three of the four variable coefficients a(x), b(x), c(x), p(x) of such equations can be chosen arbitrarily, and the remaining coefficient can be expressed through the others. Examples of specific equations and their exact solutions are given. The results obtained are generalized to more complex multidimensional nonlinear reaction–diffusion equations with variable coefficients. Also some functional separable solutions to nonlinear reaction–diffusion equations with delayut=uxx+a(x)f(u,w),w=u(x,t−τ),where τ > 0 is the delay time and f(u, w) is an arbitrary function of two arguments, are obtained.

Suggested Citation

  • Polyanin, Andrei D., 2019. "Functional separable solutions of nonlinear reaction–diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 282-292.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:282-292
    DOI: 10.1016/j.amc.2018.10.092
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    References listed on IDEAS

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    1. Pereira, Enrique & Suazo, Erwin & Trespalacios, Jessica, 2018. "Riccati–Ermakov systems and explicit solutions for variable coefficient reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 278-296.
    2. Bradshaw-Hajek, B.H. & Moitsheki, R.J., 2015. "Symmetry solutions for reaction–diffusion equations with spatially dependent diffusivity," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 30-38.
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    Citations

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    Cited by:

    1. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    2. Andrei D. Polyanin & Alexander V. Aksenov, 2024. "Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions," Mathematics, MDPI, vol. 12(13), pages 1-29, July.
    3. Andrei D. Polyanin, 2020. "Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations," Mathematics, MDPI, vol. 8(1), pages 1-38, January.
    4. Andrei D. Polyanin, 2019. "Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions," Mathematics, MDPI, vol. 7(5), pages 1-19, April.
    5. Kudryashov, Nikolay A., 2019. "Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 323-330.
    6. Yüzbaşı, Şuayip & Yıldırım, Gamze, 2022. "A collocation method to solve the parabolic-type partial integro-differential equations via Pell–Lucas polynomials," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    7. Jian Zhao & Zhenyue Chen & Jingqi Tu & Yunmei Zhao & Yiqun Dong, 2022. "Application of LSTM Approach for Predicting the Fission Swelling Behavior within a CERCER Composite Fuel," Energies, MDPI, vol. 15(23), pages 1-14, November.
    8. Andrei D. Polyanin & Vsevolod G. Sorokin, 2021. "Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy," Mathematics, MDPI, vol. 9(5), pages 1-22, March.
    9. Andrei D. Polyanin & Alexei I. Zhurov, 2022. "Multi-Parameter Reaction–Diffusion Systems with Quadratic Nonlinearity and Delays: New Exact Solutions in Elementary Functions," Mathematics, MDPI, vol. 10(9), pages 1-28, May.
    10. Cristian Ghiu & Constantin Udriste, 2022. "Solutions for Multitime Reaction–Diffusion PDE," Mathematics, MDPI, vol. 10(19), pages 1-12, October.
    11. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays," Mathematics, MDPI, vol. 11(3), pages 1-25, January.
    12. Alexander V. Aksenov & Andrei D. Polyanin, 2021. "Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions," Mathematics, MDPI, vol. 9(4), pages 1-31, February.

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