[Paper Review] Analytical Approximate Solutions of Systems of Multi-pantograph Delay Differential Equations Using Residual Power-series Method

Iryna Komashynska, Mohammed Al‐Smadi+2 more|arXiv (Cornell University)|Nov 16, 2016

Brake Systems and Friction Analysis37 references14 citations

TL;DR

This paper proposes the residual power series method (RPSM) to compute analytical approximate solutions for systems of multi-pantograph delay differential equations. The method constructs convergent Taylor-series expansions without linearization or perturbation, yielding highly accurate solutions with few iterations, demonstrating effectiveness and simplicity in numerical examples.

ABSTRACT

This paper investigates analytical approximate solutions for a system of multipantograph delay differential equations using the residual power series method (RPSM), which obtains a Taylor expansion of the solutions and produces the exact form in terms of convergent series requires no linearization or small perturbation when the solutions are polynomials. By this method, an excellent approximate solution can be obtained with only a few iterations. In this sense, computational results of some examples are presented to demonstrate the viability, simplicity and practical usefulness of the method. In addition, the results reveal that the proposed method is very effective, straightforward, and convenient for solving a system of multi-pantograph delay differential equations.

Motivation & Objective

• To develop a reliable analytical method for solving systems of multi-pantograph delay differential equations.
• To avoid the need for linearization or small perturbation techniques commonly used in perturbation-based methods.
• To provide convergent Taylor-series solutions through a systematic residual-based approach.
• To demonstrate the method's efficiency and accuracy using numerical examples.
• To establish the RPSM as a practical alternative for solving complex delay differential systems.

Proposed method

• The residual power series method constructs a Taylor series expansion of the solution around the initial point using residual functions.
• The method defines a residual function based on the original system, which is then expanded into a power series and set to zero at each order to determine coefficients.
• Coefficients of the series are computed iteratively by enforcing the residual to vanish at each order, ensuring convergence.
• The approach does not require linearization, discretization, or perturbation, preserving the original problem structure.
• The solution is expressed as a convergent power series, with higher-order terms improving accuracy.
• The method is applied iteratively to obtain successive approximations with minimal computational effort.

Experimental results

Research questions

• RQ1 Can the residual power series method provide accurate analytical approximate solutions for systems of multi-pantograph delay differential equations?
• RQ2 How does the RPSM compare to traditional methods in terms of computational efficiency and accuracy?
• RQ3 To what extent can the RPSM handle nonlinear and multi-delay systems without linearization?
• RQ4 What is the convergence behavior of the RPSM for such systems?
• RQ5 Can the method be implemented with few iterations to yield high-accuracy results?

Key findings

• The RPSM produces highly accurate analytical approximate solutions with only a few iterations, demonstrating fast convergence.
• The method does not require linearization or small perturbation, making it suitable for nonlinear systems.
• Numerical results show that the RPSM yields solutions that closely match exact solutions where available.
• The method is effective even for systems with multiple pantograph delays and complex nonlinearities.
• The residual-based iterative process ensures stability and reliability in computing series coefficients.
• The approach is straightforward to implement and computationally efficient, as demonstrated by the examples in the study.