The approximate solution of charged particle motion equations in oscillating magnetic fields using the local multiquadrics collocation method

Abstract

The charged particle motion for certain configurations of oscillating magnetic fields can be simulated by a Volterra integro-differential equation of the second order with time-periodic coefficients. This paper investigates a simple and accurate scheme for computationally solving these types of integro-differential equations. To start the method, we first reduce the integro-differential equations to equivalent Volterra integral equations of the second kind. Subsequently, the solution of the mentioned Volterra integral equations is estimated by the collocation method based on the local multiquadrics formulated on scattered points. We also expand the proposed method to solve fractional integro-differential equations including non-integer order derivatives. Since the offered method does not need any mesh generations on the solution domain, it can be recognized as a meshless method. To demonstrate the reliability and efficiency of the new technique, several illustrative examples are given. Moreover, the numerical results confirm that the method developed in the current paper in comparison with the method based on the globally supported multiquadrics has much lesser volume computing.

This is a preview of subscription content, access via your institution.

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

References

  1. 1.

    Dehghan M, Shakeri F (2008) Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method. Prog Electromagn Res 78:361–376

    Google Scholar 

  2. 2.

    Maleknejad K, Hadizadeh M, Attary M (2013) On the approximate solution of integro-differential equations arising in oscillating magnetic fields. Appl Math 58(5):595–607

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Machado JM, Tsuchida M (2002) Solutions for a class of integro-differential equations with time periodic coefficients. Appl Math E-Notes 2:66–71

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Wazwaz AM (2011) Linear and nonlinear integral equations: methods and applications. Springer, Heidelberg

    Google Scholar 

  5. 5.

    Pathak M, Joshi P (2014) High order numerical solution of a Volterra integro-differential equation arising in oscillating magnetic fields using variational iteration method. Int J Adv Sci Tech 69:47–56

    Google Scholar 

  6. 6.

    Brunner H, Makroglou A, Miller RK (1997) Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution. Appl Numer Math 23(4):381–402

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Li F, Yan T, Su L (2014) Solution of an integral-differential equation arising in oscillating magnetic fields using local polynomial regression. Adv Mech Eng 1–9:2014

    Google Scholar 

  8. 8.

    Khan Y, Ghasemi M, Vahdati S, Fardi M (2014) Legendre multi-wavelets to solve oscillating magnetic fields integro-differential equations. UPB Sci Bull Ser A 76(1):51–58

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Parand K, Rad JA (2012) Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions. Appl Math Comput 218(9):5292–5309

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Ghasemi M (2014) Numerical technique for integro-differential equations arising in oscillating magnetic fields. Iran J Sci Technol A 38(4):473–479

    MathSciNet  Google Scholar 

  11. 11.

    Assari P (2018) The thin plate spline collocation method for solving integrodifferential equations arisen from the charged particle motion in oscillating magnetic fields. Eng Comput 34:1706–1726

    Google Scholar 

  12. 12.

    Assari P, Dehghan M (2018) Solving a class of nonlinear boundary integral equations based on the meshless local discrete Galerkin (MLDG) method. Mediterr J Math 15:1–21

    MATH  Google Scholar 

  13. 13.

    Drozdov AD, Gil MI (1996) Stability of a linear integro-differential equation with periodic coefficients. Q Appl Math 54(4):609–624

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Hardy RL (2006) Hardy, multiquadric equations of topography and other irregular surfaces. J Geophys Res 176(8):1905–1915

    Google Scholar 

  15. 15.

    Fu Z, Chen W, Chen CS (2014) Recent advances in radial basis function collocation methods. Springer, New York

    Google Scholar 

  16. 16.

    Kansa EJ (1990) Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics-I. Comput Math Appl 19:127–145

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Kansa EJ (1990) Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics-II. Comput Math Appl 19:147–161

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Fu Z, Reutskiy S, Sun H, Ma J, Khan MA (2019) A robust kernel-based solver for variable-order time fractional PDEs under 2D/3D irregular domains. Appl Math Lett 94:105–111

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Fu Z, Xi Q, Chen W, Cheng AH-D (2018) A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations. Comput Math Appl 76(4):760–773

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Wendland H (2005) Scattered data approximation. Cambridge University Press, New York

    Google Scholar 

  21. 21.

    Lee CK, Liu X, Fan SC (2003) Local multiquadric approximation for solving boundary value problems. Comput Mech 30(5–6):396–409

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Sarler B, Vertnik R (2006) Meshfree explicit local radial basis function collocation method for diffusion problems. Comput Math Appl 51(8):1269–1282

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Sarra SA (2012) A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains. Appl Math Comput 218(19):9853–9865

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Vertnik R, Sarler B (2006) Meshless local radial basis function collocation method for convective–diffusive solid–liquid phase change problems. Int J Numer Methods Heat Fluid Flow 16(5):617–640

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Kosec G, Sarler B (2013) Solution of a low prandtl number natural convection benchmark by a local meshless method. Int J Numer Methods Heat Fluid Flow 23(1):189–204

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Mramor K, Vertnik R, Sarler B (2013) Simulation of natural convection influenced by magnetic field with explicit local radial basis function collocation method. CMES Comput Model Eng Sci 92(4):327–352

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Hon Y, Sarler B, Yun D (2015) Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng Anal Bound Elem 57:2–8

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Siraj-Ul-Islam, Vertnik R, Sarler B (2013) Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations. Appl Numer Math 67:136–151

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Wang B (2015) A local meshless method based on moving least squares and local radial basis functions. Eng Anal Bound Elem 50:395–401

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Siraj ul Islam, Sarler B, Vertnik R, Kosec G (2012) Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled burgers’ equations. Appl Math Model 36(3):1148–1160

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Yun DF, Hon YC (2016) Improved localized radial basis function collocation method for multi-dimensional convection-dominated problems. Eng Anal Bound Elem 67:63–80

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Shu C, Ding H, Yeo KS (2003) Local radial basis funcion-based differential quadrature method and its application to solve two-dimensional incompressible navier–stokes equations. Comput Methods Appl Mech Eng 192(7–8):941–954

    MATH  Google Scholar 

  33. 33.

    Yao G, Sarler B, Chen CS (2011) A comparison of three explicit local meshless methods using radial basis functions. Eng Anal Bound Elem 35(3):600–609

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Yao G, Duo J, Chen CS, Shen LH (2015) Implicit local radial basis function interpolations based on function values. Appl Math Comput 265:91–102

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Dehghan M, Nikpour A (2013) Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method. Appl Math Model 37(18–19):8578–8599

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Sun J, Yi H, Tan H (2016) Local radial basis function meshless scheme for vector radiative transfer in participating media with randomly oriented axisymmetric particles. Appl Opt 55(6):1232–1240

    Google Scholar 

  37. 37.

    Mavric B, Sarler B (2015) Local radial basis function collocation method for linear thermoelasticity in two dimensions. Int J Numer Methods Heat Fluid 25(6):1488–1510

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Dehghan M, Abbaszadeh M (2017) The meshless local collocation method for solving multi-dimensional Cahn–Hilliard, swift-Hohenberg and phase field crystal equations. Eng Anal Bound Elem 78:49–64

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Dehghan M, Abbaszadeh M (2016) The space-splitting idea combined with local radial basis function meshless approach to simulate conservation laws equations. Alex Eng J. https://doi.org/10.1016/j.aej.2017.02.024

    Article  Google Scholar 

  40. 40.

    Assari P, Adibi H, Dehghan M (2013) A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis. J Comput Appl Math 239(1):72–92

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Assari P, Dehghan M (2017) A meshless method for the numerical solution of nonlinear weakly singular integral equations using radial basis functions. Eur Phys J Plus 132:1–23

    MATH  Google Scholar 

  42. 42.

    Assari P, Dehghan M (2017) A meshless discrete collocation method for the numerical solution of singular-logarithmic boundary integral equations utilizing radial basis functions. Appl Math Comput 315:424–444

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Assari P, Adibi H, Dehghan M (2014) The numerical solution of weakly singular integral equations based on the meshless product integration (MPI) method with error analysis. Appl Numer Math 81:76–93

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Mirzaei D, Dehghan M (2010) A meshless based method for solution of integral equations. Appl Numer Math 60(3):245–262

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Dehghan M, Salehi R (2012) The numerical solution of the non-linear integro-differential equations based on the meshless method. J Comput Appl Math 236(9):2367–2377

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Li X (2011) Meshless Galerkin algorithms for boundary integral equations with moving least square approximations. Appl Numer Math 61(12):1237–1256

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Li X, Zhu J (2009) A meshless Galerkin method for stokes problems using boundary integral equations. Comput Methods Appl Mech Eng 198:2874–2885

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Fu Z, Chen W, Ling L (2015) Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Eng Anal Bound Elem 57:37–46

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Fu Z, Chen W, Yang H (2013) Boundary particle method for laplace transformed time fractional diffusion equations. J Comput Phys 235:52–66

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Arqub OA, Al-Smadi M, Shawagfeh N (2013) Solving Fredholm integro-differential equations using reproducing Kernel Hilbert space method. Appl Math Comput 219(17):8938–8948

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Arqub OA, Al-Smadi M (2014) Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations. Appl Math Comput 243(15):911–922

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Shawagfeh N, Arqub OA, Momani SM (2014) Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method. J Comput Anal Appl 16(4):750–762

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Halliday D, Resnick R, Walker J (1997) Fundamentals of physics. Willey, Hoboken

    Google Scholar 

  54. 54.

    Harrington RF (2003) Introduction to electromagnetic engineering. Courier Corporation

  55. 55.

    Sadiku MNO (2007) Elements of electromagnetics. Oxford University Press, Oxford

    Google Scholar 

  56. 56.

    Bojeldain AA (1991) On the numerical solving of nonlinear Volterra integro-differential equations. Ann Univ Sci Bp Sect Comput 11:105–125

    MathSciNet  MATH  Google Scholar 

  57. 57.

    Fu Z, Chen W, Wen P, Zhang C (2018) Singular boundary method for wave propagation analysis in periodic structures. J Sound Vib 425:170–188

    Google Scholar 

  58. 58.

    Fasshauer GE (2005) Meshfree methods. In Handbook of theoretical and computational nanotechnology, American Scientific Publishers

  59. 59.

    Assari P, Asadi-Mehregan F (2019) Local multiquadric scheme for solving two-dimensional weakly singular Hammerstein integral equations. Int J Numer Model 32(1):1–23

    MATH  Google Scholar 

  60. 60.

    Buhmann MD (2003) Radial basis functions: theory and implementations. Cambridge University Press, Cambridge

    Google Scholar 

  61. 61.

    Quarteroni A, Sacco R, Saleri F (2008) Numerical analysis for electromagnetic integral equations. Artech House, Boston

    Google Scholar 

  62. 62.

    Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge University Press, Cambridge

    Google Scholar 

  63. 63.

    Zhang S, Lin Y, Rao M (2000) Numerical solutions for second-kind Volterra integral equations by Galerkin methods. Appl Math 45(1):19–39

    MathSciNet  MATH  Google Scholar 

  64. 64.

    Arqub OA, Maayah B (2018) Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator. Chaos Solitons Fractals 117:117–124

    MathSciNet  MATH  Google Scholar 

  65. 65.

    Arqub OA, Al-Smadi M (2018) Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painleve equations in Hilbert space. Chaos Solitons Fractals 117:161–167

    MathSciNet  MATH  Google Scholar 

  66. 66.

    Arqub OA, Maayah B (2018) Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions. Numer Methods Partial Differ Equ 34:1577–1597

    MathSciNet  MATH  Google Scholar 

  67. 67.

    Arqub OA (2018) Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space. Numer Methods Partial Differ Equ 34:1759–1780

    MathSciNet  MATH  Google Scholar 

  68. 68.

    Arqub OA (2019) Application of residual power series method for the solution of time-fractional Schrodinger equations in one-dimensional space. Fundam Inform 166:87–110

    MathSciNet  MATH  Google Scholar 

  69. 69.

    Kaneko H, Xu Y (1994) Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Math Comput 62(206):739–753

    MathSciNet  MATH  Google Scholar 

  70. 70.

    Assari P, Asadi-Mehregan F, Dehghan M (2018) On the numerical solution of Fredholm integral equations utilizing the local radial basis function method. Int J Comput Math. https://doi.org/10.1080/00207160.2018.1500693

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to both anonymous reviewers for their valuable comments and suggestions which have improved the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pouria Assari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Assari, P., Asadi-Mehregan, F. The approximate solution of charged particle motion equations in oscillating magnetic fields using the local multiquadrics collocation method. Engineering with Computers 37, 21–38 (2021). https://doi.org/10.1007/s00366-019-00807-z

Download citation

Keywords

  • Charged particle motion equation
  • Oscillating magnetic field
  • Integro-differential equation
  • Fractional order
  • Discrete collocation method
  • Local multiquadrics

Mathematics Subject Classification

  • 83C50
  • 45D05
  • 45J05