Saeid Abbasbandy

We present a hybrid computational framework that integrates quantum representations by helping tensor products with classical spectral methods to solve pantograph equations, a prominent class of delay-differential equations with applications across applied mathematics. Both single and multiple delay cases are considered. Our approach establishes a Chebyshev-based spectral framework, leveraging careful selection of basis functions and collocation points. Nonlinearities are addressed via successive linearization, which iteratively reduces the problem to systems of linear equations solved with a quantum algorithm. The solution is then encoded as a quantum state, enabling efficient processing. The methodology is validated through numerical experiments, with comparative results confirming its efficiency. The numerical results presented in this work were obtained using classical computation. The quantum formalism serves as a blueprint for potential future implementation on quantum hardware.

This paper introduces a new technique utilizing the reproducing kernel method (RKM) to solve both linear and nonlinear systems of fractional‐order differential equations (SFDEs). The technique carefully integrates essential elements, including the solution space, basis functions, strategic point selection, and a suitable inner product. While solving SFDEs can be notoriously complex, leading to issues such as extended computation times, elevated matrix condition numbers, reduced accuracy compared to alternative methods, and discrepancies between theoretical predictions and numerical outcomes, our approach effectively mitigates these challenges. We achieve this by eliminating the need for Gram–Schmidt orthogonalization and incorporating efficient linear algebra techniques. By strategically selecting the solution space and evaluation points, we develop an RKM‐based approach that combines computational efficiency, high accuracy, and straightforward implementation.

The Painlevé equations and their series and rational solutions are essential in applied, pure mathematics and theoretical physics. Recently, quantum algorithms have helped to implement numerical algorithms more easily by performing linear algebra in our working. This article uses a hybrid of quantum computing schemes and spectral methods for the second Painlevé equation. Two approaches are investigated: first, a series solution is obtained, and then the rational solutions. The successive linearization method is used for the linearization of the Painlevé‐II equation. In the computer implementation, the solution value of the Painlevé‐II equation is considered as a final quantum state. In each iterative scheme, by adding the current quantum state, we can compute the final quantum state. We need different quantum models for each approach, series, and rational solutions. Numerical examples illustrate the efficiency of this method, and reasonable solutions are obtained for a wide range of parameter values.

In this paper, we introduce a new approach based on the Reproducing Kernel Method (RKM) for solving a nonlinear system of second-order Boundary Value Problems (BVPs) without the Gram-Schmidt orthogonalization process. What motivates us to use the RKM without the Gram-Schmidt orthogonalization process is its easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy. Finally, the compatibility of numerical results and theorems demonstrates that the Present method is effective.

In this article, a new implementation of the reproducing kernel method is presented for solving systems of fractional-order Volterra integro-differential equations. Unlike previous implementations, this method does not rely on the Gram-Schmidt process. The reproducing kernel method utilizes various components, including space, inner product, bases, and points. Furthermore, the system of fractional-order Volterra integro-differential equations involves Caputo’s fractional derivative and Volterra integral. However, when using the reproducing kernel method to solve these systems, challenges such as longer execution time and lower accuracy may arise compared to other methods. The present method has overcome these challenges with features such as easy implementation, high accuracy, and lower execution time.