Solving the Integral Differential Equations with Delayed Argument by Using the DTM Method.

Area of Science:

• Applied Mathematics
• Computer Science
• Engineering

Background:

• Wireless sensor networks and the Industrial Internet of Things (IIoT) are rapidly advancing.
• Practical applications in smart industries, devices, and transportation necessitate advanced mathematical solutions.
• Existing methods struggle with the complexity of differential, integral, and integro-differential equations.

Purpose of the Study:

• To present an efficient method for solving integro-differential equations with retarded arguments.
• To adapt advanced mathematical techniques for IIoT and wireless sensor network challenges.
• To simplify complex equations into solvable algebraic systems.

Main Methods:

• The study employs the Taylor differential transformation method.
• This technique converts integro-differential equations into systems of algebraic equations.
• The method is designed for equations with delayed arguments.

Main Results:

• The Taylor differential transformation method effectively transforms complex integro-differential equations.
• The transformation results in a system of nonlinear algebraic equations.
• The method demonstrates efficiency and relative simplicity in application.

Conclusions:

• The proposed method offers a valuable approach for tackling complex mathematical problems in emerging technologies.
• While a general solution form is not achievable due to problem complexity, the method provides significant benefits.
• This technique facilitates practical solutions for wireless sensor networks and IIoT applications.