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Approximating non linear higher order ODEs by a three point block algorithm.

Ahmad Fadly Nurullah Rasedee1, Mohammad Hasan Abdul Sathar2, Khairil Iskandar Othman3

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This study introduces a new numerical method, the three point block backward differentiation formula (3PBBDF), for solving ordinary differential equations (ODEs). The method enhances computational efficiency while maintaining accuracy for complex ODE models.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Background:

  • Differential equations are crucial for modeling real-world phenomena.
  • Analytical solutions for complex differential equations are often unattainable.
  • Numerical methods provide approximations but require continuous refinement for accuracy and efficiency.

Purpose of the Study:

  • To develop an efficient numerical algorithm for solving higher-order ordinary differential equations (ODEs).
  • To introduce a novel three-point block multistep method based on Adams-type formulas (3PBCS).
  • To evaluate the proposed method's accuracy and computational efficiency.

Main Methods:

  • Development of a three-point block multistep method (3PBCS).
  • Application of the method to various types of ODEs: linear, nonlinear, artificial, and real-life problems.
  • Analysis of the method's order, stability, and convergence properties.

Main Results:

  • The proposed 3PBCS method demonstrates competitive accuracy compared to existing numerical techniques.
  • The block approach significantly enhances computational efficiency by reducing the number of function evaluations.
  • The method is shown to be stable and convergent for the tested ODEs.

Conclusions:

  • The 3PBCS method offers a viable and efficient alternative for solving higher-order ODEs.
  • This research contributes to the advancement of numerical methods for differential equations.
  • The findings highlight the importance of balancing accuracy and computational cost in numerical algorithm design.