Skip to main content
Springer Nature Link
Log in

Local integration of population dynamics via moving least squares approximation

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

This paper applies an approach based on the Galerkin and collocation methods so-called meshless local Petrov–Galerkin (MLPG) method to treat a nonlinear partial integro-differential equation arising in population dynamics. In the proposed method, the MLPG method is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The moving least squares approximation is proposed to construct shape functions. A one-step time discretization method is employed to approximate the time derivative. To treat the nonlinearity, a simple predictor–corrector scheme is performed. Also the integral term, which is a kind of convolution, is treated by the cubic spline interpolation. Convergence in both time and spatial discretizations is shown and more, stability of the method is illustrated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Apreutesei N, Bessonov N, Volpert V, Vougalter V (2010) Spatial structures and generalized travelling waves for an integro-differential equation. Discrete Contin Dyn Syst Ser B 13:537–557

    Article  MathSciNet  MATH  Google Scholar 

  2. Shivanian E (2013) Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics. Eng Anal Bound Elem 37:1693–1702

    Article  MathSciNet  MATH  Google Scholar 

  3. Fakhar-Izadi F, Dehghan M (2013) An efficient pseudo-spectral Legendre–Galerkin method for solving a nonlinear partial integro-differential equation arising in population dynamics. Math Methods Appl Sci 36(12):1485–1511

    Article  MathSciNet  MATH  Google Scholar 

  4. Ǵenieys S, Volpert V, Auger P (2006) Pattern and waves for a model in population dynamics with nonlocal consumption of resources. Math Model Nat Phenom 1:65–82

    MathSciNet  MATH  Google Scholar 

  5. Britton NF (1986) Aggregation and the competitive exclusion principle. J Theor Biol 136:57–66

    Article  MathSciNet  Google Scholar 

  6. Britton NF (1990) Spatial structures and periodic travelling waves in an integro-differential reaction–diffusion population model. SIAM J Appl Math 50:1663–1688

    Article  MathSciNet  MATH  Google Scholar 

  7. Al-Khaled K (2001) Numerical study of Fisher’s reaction–diffusion equation by the sinc collocation method. J Comput Appl Math 137:245–255

    Article  MathSciNet  MATH  Google Scholar 

  8. Branco JR, Ferrèira JA, de Oliveira P (2007) Numerical methods for the generalized Fisher–Kolmogorov–Piskunov equation. Appl Numer Math 57:89–102

    Article  MathSciNet  MATH  Google Scholar 

  9. Carey GF, Shen Y (1995) Least-squares fnite element approximation of Fisher’s reaction–diffusion equation. Numer Methods Partial Differ Equ 11:175–186

    Article  MathSciNet  MATH  Google Scholar 

  10. Daǧ I, Şahin A, Korkmaz A (2010) Numerical investigation of the solution of Fisher’s equation via the B-spline Galerkin method. Numer Methods Partial Differ Equ 26:1483–1503

    MATH  Google Scholar 

  11. Ǵenieys S, Bessonov N, Volpert V (2009) Mathematical model of evolutionary branching. Math Comput Model 49:2109–2115

    Article  MathSciNet  MATH  Google Scholar 

  12. Khuri SA, Sayfy A (2010) A numerical approach for solving an extended Fisher–Kolomogrov–Petrovskii–Piskunov equation. J Comput Appl Math 233:2081–2089

    Article  MathSciNet  MATH  Google Scholar 

  13. Mickens RE (1994) A best finite-difference scheme for Fisher’s equation. Numer Methods Partial Differ Equ 10:581–585

    Article  MathSciNet  MATH  Google Scholar 

  14. Olmos D, Shizgal B (2006) A spectral method of solution of Fisher’s equation. J Comput Appl Math 193:219–242

    Article  MathSciNet  MATH  Google Scholar 

  15. Perthame B, Ǵenieys S (2007) Concentration in the nonlocal fisher equation: the Hamilton-Jacobi limit. Math Model Nat Phenom 4:135–151

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu G, Gu Y (2005) An introduction to meshfree methods and their programing. Springer, Berlin

    Google Scholar 

  17. Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229–256

    Article  MathSciNet  MATH  Google Scholar 

  18. Belytschko T, Lu YY, Gu L (1995) Element free Galerkin methods for static and dynamic fracture. Int J Solids Struct 32:2547–2570

    Article  MATH  Google Scholar 

  19. Dehghan M, Ghesmati A (2010) Combination of meshless local weak and strong (mlws) forms to solve the two dimensional hyperbolic telegraph equation. Eng Anal Bound Elem 34(4):324–336

    Article  MathSciNet  MATH  Google Scholar 

  20. Kansa E (1990) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates. Comput Math Appl 19(8–9):127–145

    Article  MathSciNet  MATH  Google Scholar 

  21. Dehghan M, Shokri A (2008) A numerical method for solution of the twodimensional sine-Gordon equation using the radial basis functions. Math Comput Simul 79:700–715

    Article  MathSciNet  MATH  Google Scholar 

  22. Pan R, Skala V (2011) A two-level approach to implicit surface modeling with compactly supported radial basis functions. Eng Comput 27(3):299–307

    Article  Google Scholar 

  23. Dehghan M, Shokri A (2009) Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions. J Comput Appl Math 230(2):400–410

    Article  MathSciNet  MATH  Google Scholar 

  24. Aslefallah M, Shivanian E (2015) Nonlinear fractional integro-differential reaction–diffusion equation via radial basis functions. Eur Phys J Plus 130(47):1–9

    Google Scholar 

  25. Shivanian E (2015) A meshless method based on radial basis and spline interpolation for 2-D and 3-D inhomogeneous biharmonic BVPs. Zeitschrift für Naturforschung A 70(8):673–682

    Article  Google Scholar 

  26. Ling L, Opfer R, Schaback R (2006) Results on meshless collocation techniques. Eng Anal Bound Elem 30(4):247–253

    Article  MATH  Google Scholar 

  27. Ling L, Schaback R (2008) Stable and convergent unsymmetric meshless collocation methods. SIAM J Numer Anal 46(3):1097–1115

    Article  MathSciNet  MATH  Google Scholar 

  28. Libre N, Emdadi A, Kansa E, Shekarchi M, Rahimian M (2008) A fast adaptive wavelet scheme in RBF collocation for nearly singular potential PDEs. CMES-Comput Model Eng Sci 38(3):263–284

    MathSciNet  MATH  Google Scholar 

  29. Libre N, Emdadi A, Kansa E, Shekarchi M, Rahimian M (2009) A multiresolution prewavelet-based adaptive refinement scheme for RBF approximations of nearly singular problems. Eng Anal Bound Elem 33(7):901–914

    Article  MathSciNet  MATH  Google Scholar 

  30. Fakhar-Izadi F, Dehghan M (2014) Space-time spectral method for a weakly singular parabolic partial integro-differential equation on irregular domains. Comput Math Appl 67(10):1884–1904

    Article  MathSciNet  Google Scholar 

  31. Fatahi H, Saberi-Nadjafi J, Shivanian E (2016) A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional fredholm integral equations on general domains with error analysis. J Comput Appl Math 294:196–209

    Article  MathSciNet  MATH  Google Scholar 

  32. Shivanian E (2015) Spectral meshless radial point interpolation (SMRPI) method to two-dimensional fractional telegraph equation. Math Methods Appl Sci. doi:10.1002/mma.3604

    Google Scholar 

  33. Atluri S, Zhu T (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117–127

    Article  MathSciNet  MATH  Google Scholar 

  34. Atluri S, Zhu T (1998) A new meshless local Petrov–Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation. Comput Model Simul Eng 3(3):187–196

    Google Scholar 

  35. Atluri S, Zhu T (2000) New concepts in meshless methods. Int J Numer Methods Eng 13:537–556

    Article  MathSciNet  MATH  Google Scholar 

  36. Atluri S, Zhu T (2000) The meshless local Petrov–Galerkin (MLPG) approach for solving problems in elasto-statics. Comput Mech 25:169–179

    Article  MATH  Google Scholar 

  37. Dehghan M, Mirzaei D (2008) The meshless local Petrov–Galerkin (MLPG) method for the generalized two-dimensional non-linear schrödinger equation. Eng Anal Bound Elem 32:747–756

    Article  MATH  Google Scholar 

  38. Dehghan M, Mirzaei D (2009) Meshless local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity. Appl Numer Math 59:1043–1058

    Article  MathSciNet  MATH  Google Scholar 

  39. Gu Y, Liu G (2001) A meshless local Petrov–Galerkin (MLPG) method for free and forced vibration analyses for solids. Comput Mech 27:188–198

    Article  MATH  Google Scholar 

  40. Abbasbandy S, Shirzadi A (2010) A meshless method for two-dimensional diffusion equation with an integral condition. Eng Anal Bound Elem 34(12):1031–1037

    Article  MathSciNet  MATH  Google Scholar 

  41. Shivanian E (2015) Meshless local Petrov–Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation. Eng Anal Bound Elem 50:249–257

    Article  MathSciNet  Google Scholar 

  42. Shivanian E, Abbasbandy S, Alhuthali MS, Alsulami HH (2015) Local integration of 2-D fractional telegraph equation via moving least squares approximation. Eng Anal Bound Elem 56:98–105

    Article  MathSciNet  Google Scholar 

  43. Dehghan M, Salehi R (2014) A meshless local Petrov–Galerkin method for the time-dependent maxwell equations. J Comput Appl Math 268:93–110

    Article  MathSciNet  MATH  Google Scholar 

  44. Taleei A, Dehghan M (2014) Direct meshless local Petrov–Glerkin method for elliptic interface problems with applications in electrostatic and elastostatic. Comput Methods Appl Mech Eng 278:479–498

    Article  MathSciNet  Google Scholar 

  45. Dehghan M, Abbaszadeh M, Mohebbi A (2014) Numerical solution of system of n-coupled nonlinear schrodinger equations via two variants of the meshless local Petrov–Galerkin (mlpg) method. Comput Model Eng Sci CMES 100(5):399–444

    MathSciNet  Google Scholar 

  46. Salehi R, Dehghan M (2013) A moving least square reproducing polynomial meshless method. Appl Numer Math 69:34–58

    Article  MathSciNet  MATH  Google Scholar 

  47. Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318

    Article  MATH  Google Scholar 

  48. Bratsos A (2008) An improved numerical scheme for the sine-Gordon equation in 2+1 dimensions. Int J Numer Meth Eng 75:787–799

    Article  MathSciNet  MATH  Google Scholar 

  49. Clear P (1998) Modeling conned multi-material heat and mass flows using SPH. Appl Math Model 22:981–993

    Article  Google Scholar 

  50. Liu W, Jun S, Zhang Y (1995) Reproducing kernel particle methods. Int J Numer Methods Eng 20:1081–1106

    Article  MathSciNet  MATH  Google Scholar 

  51. Mukherjee Y, Mukherjee S (1997) Boundary node method for potential problems. Int J Numer Methods Eng 40:797–815

    Article  MATH  Google Scholar 

  52. Melenk J, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314

    Article  MathSciNet  MATH  Google Scholar 

  53. De S, Bathe K (2000) The method of finite spheres. Comput Mech 25:329–345

    Article  MathSciNet  MATH  Google Scholar 

  54. Gu Y, Liu G (2002) A boundary point interpolation method for stress analysis of solids. Comput Mech 28:47–54

    Article  MATH  Google Scholar 

  55. Gu Y, Liu G (2003) A boundary radial point interpolation method (BRPIM) for 2-d structural analyses. Struct Eng Mech 15:535–550

    Article  Google Scholar 

  56. Liu G, Yan L, Wang J, Gu Y (2002) Point interpolation method based on local residual formulation using radial basis functions. Struct Eng Mech 14:713–732

    Article  Google Scholar 

  57. Liu G, Gu Y (2001) A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids. J Sound Vib 246(1):29–46

    Article  Google Scholar 

  58. Dehghan M, Ghesmati A (2010) Numerical simulation of two-dimensional sine-gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Comput Phys Commun 181:772–786

    Article  MathSciNet  MATH  Google Scholar 

  59. Shivanian E, Khodabandehlo H (2014) Meshless local radial point interpolation (MLRPI) on the telegraph equation with purely integral conditions. Eur Phys J Plus 129:241–251

    Article  Google Scholar 

  60. Hosseini V, Shivanian E, Chen W (2015) Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation. Eur Phys J Plus 130:33–54

    Article  Google Scholar 

  61. Shivanian E (2014) Analysis of meshless local and spectral meshless radial point interpolation (MLRPI and SMRPI) on 3-D nonlinear wave equations. Ocean Eng 89:173–188

    Article  Google Scholar 

  62. Shivanian E (2015) A new spectral meshless radial point interpolation (SMRPI) method: a well-behaved alternative to the meshless weak forms. Eng Anal Bound Elem 54:1–12

    Article  MathSciNet  Google Scholar 

  63. Shivanian E On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three-dimensional wave equations. Int J Numer Methods Eng (in press). doi:10.1002/nme.4960

  64. Abbasbandy SE (2015) The effects of MHD flow of third grade fluid by means of meshless local radial point interpolation (MLRPI). Int J Ind Math 7(1):1–11

    Google Scholar 

  65. Shivanian E, Rahimi A, Hosseini M (2015) Meshless local radial point interpolation to three-dimensional wave equation with neumann’s boundary conditions. Int J Comput Math. doi:10.1080/00207160.2015.1085032

    Google Scholar 

  66. Shivanian E, Khodabandehlo HR (2015) Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem. Ain Shams Eng J. doi:10.1016/j.asej.2015.07.009

    Google Scholar 

  67. Lei Z, Tianqi G, Ji Z, Shijun J, Qingzhou S, Ming H (2014) An adaptive moving total least squares method for curve fitting. Measurement 49:107–112

    Article  Google Scholar 

  68. Hu D, Long S, Liu K, Li G (2006) A modified meshless local Petrov–Galerkin method to elasticity problems in computer modeling and simulation. Eng Anal Bound Elem 30:399–404

    Article  MATH  Google Scholar 

  69. Liu K, Long S, Li G (2006) A simple and less-costly meshless local Petrov–Galerkin (MLPG) method for the dynamic fracture problem. Eng Anal Bound Elem 30:72–76

    Article  MATH  Google Scholar 

  70. de Boor C (1978) A practical guide to splines. Springer, Berlin

    Book  MATH  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to two anonymous reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper very much.

Author information

Authors and Affiliations

  1. Department of Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran

    E. Shivanian

Authors
  1. E. Shivanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Shivanian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shivanian, E. Local integration of population dynamics via moving least squares approximation. Engineering with Computers 32, 331–342 (2016). https://doi.org/10.1007/s00366-015-0424-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-015-0424-z

Keywords