A new multi-step technique with differential transform method for analytical solution of some nonlinear variable...

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Differential Equations

Background:

• Nonlinear delay differential equations (DDEs) with variable delays pose significant challenges for traditional numerical and analytical methods.
• Existing general-purpose codes often struggle to accurately solve these complex equations.

Purpose of the Study:

• To develop a robust analytical solution for nonlinear DDEs with variable delays.
• To present a novel method that overcomes the limitations of existing numerical and analytical approaches.

Main Methods:

• A new method of steps is combined with the differential transform method (DTM).
• This approach transforms DDEs into ordinary differential equations, solvable by DTM.
• Laplace-Padé resummation is employed to enhance solution accuracy.

Main Results:

• The proposed method effectively provides analytical solutions for nonlinear DDEs with variable delays.
• Demonstrated efficiency through two illustrative examples.
• The technique offers a simplified procedure adaptable to other analytical methods like homotopy perturbation method.

Conclusions:

• The combined method of steps and DTM offers a powerful and efficient tool for solving challenging nonlinear DDEs.
• The approach enhances solution accuracy and can be integrated with various analytical techniques.
• This work provides a valuable advancement in the analytical treatment of DDEs.