A backward difference formulation for analyzing the dynamics of capital stocks

The current study provides a numerical method that is derived in a backward difference formulation for ordinary differential equations.  The proposed method employs a constant step size algorithm of order 12.  The backward difference formulation serves as a competitive algorithm for solving ordinary differential equations.  In the current study, the backward difference method is used to analyze the dynamics of capital stocks in terms of depreciation rate for the capital–labor ratio.  Results provided in this study will validate the accuracy of the backward difference algorithm hence proving it as a viable alternative for analyzing  economic problems in the form of ordinary differential equations.

applied mathematics
backward difference
ODEs
multistep
  1. Krogh F. T.  A variable-step, variable-order multistep method for the numerical solution of ordinary differential equations.  Proc. of the IFIP Congress in Information Processing. 68, 194 (1968).
  2. Hall G., Watt J. M.  Modern numerical methods for ordinary differential equations.  Clarendon Press (1976).
  3. Suleiman M. B.  Generalised multistep Adams and backward differentiation methods for the solution of stiff and non-stiff ordinary differential equations.  University of Manchester PhD Thesis (1979).
  4. Omar Z.  Developing parallel block methods for solving higher order ODEs directly.  University Putra Malaysia PhD Thesis (1999).
  5. Abdul Majid Z.  Parallel block methods for solving ordinary differential equations.  University Putra Malaysia PhD Thesis (2004).
  6. Ibrahim Z. B.  Block multistep methods for solving ordinary differential equations.  University Putra Malaysia PhD Thesis (2006).
  7. Othman K. I.  Partitioning Techniques and Their Parallelization for Stiff System of Ordinary Differential Equations.  University Putra Malaysia PhD Thesis (2007).
  8. Rasedee A. F. N.  Direct method using backward difference for solving higher order ordinary differential equations.  University Putra of Malaysia Masters Thesis (2009).
  9. Rasedee A. F. N., Mohd Ijam H., Abdul Sathar M. H., Hamzah S. R., Ishak N., Sahrim M., Ismail I.  Solution for nonlinear Riccati equation by block method.  AIP Conference Proceedings. 1974 (1), 020071, (2018).
  10. Suleiman M. B., Ibrahim Z. B., Rasedee A. F. N.  Solution of higher-order ODEs using backward difference method.  Mathematical Problems in Engineering.  2011, Article ID 810324  (2011).
  11. Rasedee A. F. N., Abdul Sathar M. H., Deraman F., Mohd Ijam H., Suleiman M., Saaludin N., Rakhimov A.  2 point block backward difference method for solving Riccati type differential problems.
     AIP Conference Proceedings.  1775, 030005 (2016)    .
  12. Mohd Ijam H., Suleiman M. B., Rasedee A. F. N., Senu N., Ahmadian A., Salahshour S.  Solving nonstiff higher-order ordinary differential equations using 2-point block method directly.  Abstract and Applied Analysis. 2014, Article ID 867095 (2014).
  13. Mohd Ijam H., Ibrahim Z. B., Suleiman M. B., Senu N., Rasedee A. F. N.  Order and stability of 2-point block backward difference method.  AIP Conference Proceedings. 1974, 020054 (2018).
  14. Ibrahim Z. B., Zainuddin N., Othman K. I., Suleiman M., Zawawi I. S. M.  Variable Order Block Method for Solving Second Order Ordinary Differential Equations.  Sains Malaysiana. 48 (8), 1761–1769 (2019).
  15. Adeyeye O., Omar Z.  Implicit five-step block method with generalised equidistant points for solving fourth order linear and non-linear initial value problems.  Ain Shams Engineering Journal. 10 (4), 881–889 (2019).
  16. Asnor A. I., Mohd Yatim S. A., Ibrahim Z. B.  Solving Directly Higher Order Ordinary Differential Equations by Using Variable Order Block Backward Differentiation Formulae.  Symmetry. 11 (10), 1289 (2019).
  17. Mohd Nasir N., Abdul Majid Z., Ismail F., Bachok N.  Direct integration of the third-order two point and multipoint Robin type boundary value problems.  Mathematics and Computers in Simulation. 182 (1), 411–427 (2021).
  18. Rasedee A. F. N., Abdul Sathar M. H., Hamzah S. R., Ishak N., Wong T. Z., Koo L. F., Ibrahim S. N. I.  Block variable order step size Multistep Method for Solving Higher Order Ordinary Differential Equations Directly.  Journal of King Saud University – Science. 33 (3), 101376 (2021).
  19. Rasedee A. F. N., Abdul Sathar M. H., Othman K. I., Hamzah S. R., Ishak N.  Two-Point Approximating non linear higher order ODEs by a three point block algorithm.  Plos One. 16 (2), e0246904 (2021).
  20. Rasedee A. F. N., Suleiman M. B., Ibrahim Z. B.  Solving nonstiff higher order odes using variable order step size backward difference directly.  Mathematical Problems in Engineering. 2014, Article ID 565137 (2014).
  21. Rasedee A. F. N., Hamzah S. R., Ishak N., Mohd Ijam H., Suleiman M. B., Ibrahim Z. B., Abdul Sathar M. H., Ramli N. A., Kamaruddin N. S.  Variable order variable stepsize algorithm for solving nonlinear Duffing oscillator.  Journal of Physics: Conference Series. 890, 012045 (2017).
  22. Solow R. M.  A contribution to the theory of economic growth.  The Quarterly Journal of Economics. 70 (1), 65–94 (1956).
  23. Swan T. W.  Economic growth and capital accumulation.  Economic records. 32 (2), 334–361 (1956).
  24. Barro R. J., Sala-i-Martin X.  Economic Growth.  McGraw-Hill (2004).
  25. Zhang W. B.  Economic dynamics: growth and development.  Springer Science & Business Medial (2012).