International Scientific Journal

Thermal Science - Online First

online first only

The space spectral interpolation collocation method for reaction-diffusion systems

A space spectral interpolation collocation method is proposed to study nonlinear reaction-diffusion systems with complex dynamics characters. A detailed solution process is elucidated, and some pattern formations are given. The numerical results have a good agreement with theoretical ones. The method can be extended to fractional calculus and fractal calculus.
PAPER REVISED: 2020-06-26
PAPER ACCEPTED: 2020-06-28
  1. Qiu, Y.Y. Numerical approach to the time-fractional reaction-diffusion equation, Thermal Science, 23(2019), 4, pp. 2245-2251
  2. Cao, L. and Ma, Z.X. Numerical solution of a class of advection-reaction-diffusion system, Thermal Science, 23(2019), 3A, pp.1503-1511
  3. Dai, H.P., et al. Lattice Boltzmann model for the Riesz space fractional reaction-diffusion, Thermal Science, 22(2018), 4, pp. 1831-1843
  4. Das, S., et al. Numerical solution of fractional order advection-reaction-diffusion equation,Thermal Science, 22(2018), Supplement 1 , pp. S309-S316
  5. Bassett, A., et al. Continuous Dispersal in a Model of Predator-prey-subsidy Population Dynamics, Ecological Modelling 354(2017), pp.115-122.
  6. Cardone G., Perugia C.,Timofte C., Homogenization Results for a Coupled System of Reaction diffusion Equations, Nonlinear Analysis 188(2019)236-264.
  7. Bhowmik S. K., A Multigrid Preconditioned Numerical Scheme for a Reaction-diffusion System, Applied Mathematics and Computation 254(2015), pp.266-276.
  8. Garvie M. R., Finite-difference Schemes for Reaction-diffusion Equations Modeling Predatorprey Interactions in Matlab, Florida State University, Tallahassee (2007), pp.931-955.
  9. Medvinsky, A. B., et al., Spatiotemporal Complexity of Plankton and Fish Dynamics, Siam Review 44(2002), pp.311-370.
  10. Wang Y. M., Zhang H. B., Higher-order compact finite difference method for systems of reaction-diffusion equations, Journal of Computational and Applied Mathematics 233(2009), pp.502-518.
  11. Macha J.,et al., Nonlinear Galerkin Finite Element Method Applied to the System of Reaction-diffusion Equations in one Space Dimension, Computers and Mathematics with Applications 73(2017) , pp.2053-2065.
  12. Harizanov S., Lazarov R.,et al., Numerical Solution of Fractional Diffusion-reaction Problems Based on BURA, Computers and Mathematics with Applications (2019),
  13. He J. H., Wu X. H., Construction of Solitary Solution and Compacton-like Solution by Variational Iteration Method, Chaos Solitons Fractals 29(2006), pp.108-113.
  14. He J. H.,Variational Iteration Method-a Kind of Non-linear Analytical Technique: Some Examples, International Journal of Non-Linear Mechanics 34(1999), pp.699-708.
  15. He J. H., A Coupling Method of a Homotopy Technique and a Perturbation Technique for Nonlinear Problems, International journal of non-linear mechanics 35(2000), pp.37-43.
  16. He J. H., Homotopy Perturbation Method: A New Nonlinear Analytical Technique, Applied Mathematics and Computation 135(2003), pp.73-79.
  17. He J. H., New Interpretation of Homotopy Perturbation Method, International Journal of Modern Physics B 20(2006), pp.2561-668.
  18. Liu,F. F. , et al. Barycentric Interpolation Collocation Method for Solving the Coupled Viscous Burgers' Equations, International Journal of Computer Mathematics 95(2018), pp.2162-2173.
  19. Wu H. C., Wang Y. L., Zhang W., Numerical Solution of a Class of Nonlinear Partial Differential Equations by Using Barycentric Interpolation Collocation Method, Mathematical Problems in Engineering 2018, ID 7260346.
  20. Wu, H.C., et al. The barycentric interpolation collocation method for a class of nonlinear vibration systems, Journal of Low Frequency Noise Vibration and Active Control, 38(2019), 3-4, pp. 1504-1495
  21. Wang Y. L., Li Z. Y., A New Method for Solving a Class of Mixed Boundary Value Problems with Singular Coefficient, Applied Mathematics and Computation 217(2010), pp.2768-2772.
  22. Wang Y. L., Temuer C. L., Using Reproducing Kernel for Solving a Class of Singular Weakly Nonlinear Boundary Value Problems, International Journal of Computer Mathematics 87(2010), pp.367-380.
  23. Wang Y. L., Jia L. N., Zhang H. L., Numerical Solution for a Class of Space-time Fractional Equation in Reproducing Reproducing Kernel Space, International Journal of Computer Mathematics 96(2019), pp.2100-2111.
  24. Wang Y. L., Temuer C. L., Pang J., New Algorithm for Second-order Boundary Value Problems of Integro-differential Equation, Journal of Computational and Applied Mathematics 2009(229), pp.1-6.
  25. Wang Y. L., Du M. J., Tan F. G., Using Reproducing Kernel for Solving a Class of Fractional Partial Differential Equation with Non-classical Conditions, Applied Mathematics and Computation 219(2013), pp.5918-5925.
  26. Zhang X., Efficient Solution of Differential Equation Based on MATLAB: Spectral Method Principle and Implementation, Beijing China, Mechanical Industry Press 2015.
  27. Ain, Q. T., He, J. H. On Two-Scale Dimension and Its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
  28. He, J. H, Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
  29. He, J. H., et al.: New Promises and Future Challenges of Fractal Calculus, Thermal Science, 24(2020), 2A, pp. 659-681
  30. He, J. H., The Simpler, the Better: Analytical Methods for Non-Linear Oscillators and Fractional Oscillators, Journal Low Freq. Noise V. A., 38 (2019), 3-4, pp. 1252-1260
  31. He, J. H., A Simple Approach to 1-D Convection-Diffusion Equation and Its Fractional Modification for E-reaction Arising in Rotating Disk Electrodes, Journal of Electroanalytical Chemistry, 854 (2019), Dec.,113565
  32. He, J.-H., Latifizadeh, H., A General Numerical Algorithm for Non-Linear Differential Equations by the Variational Iteration Method, International Journal of Numerical Methods for Heat and Fluid-Flow, Online first,, 2020
  33. He, J. H., et al. A Fractal Boussinesq Equation for Non-Linear Transverse Vibration of a Nanofiber-Reinforced Concrete Pillar, Applied Mathematical Modelling, 82 (2020), June, pp. 437-448