Hind K. Al Jeaid

The pantograph equation is a basic model in the field of delay differential equations. This paper deals with an extended version of the pantograph delay equation by incorporating a variable coefficient of exponential order. At specific values of the involved parameters, the exact solution is obtained by applying the regular Maclaurin series expansion (MSE). A second approach is also applied on the current model based on a hybrid method combining the Laplace transform (LT) and the Adomian decomposition method (ADM) denoted as (LTADM). Although the MSE derives the exact solution in a straightforward manner, the LTADM determines the solution in a closed series form which is theoretically proved for convergence. Further, the accuracy of such a closed-form solution is examined through various comparisons with the exact solution. For validation, the residual errors are calculated and displayed in graphs. The results show that the solution obtained utilizing the LTADM is in full agreement with the exact solution using only a few terms of the closed-form series solution. Moreover, it is found that the residual errors tend to zero, which reflects the effectiveness of the LTADM. The present approach may merit further extension by including other types of linear delay differential equations with variable coefficients.

The Pantograph equation is a fundamental mathematical model in the field of delay differential equations. A special case of the Pantograph equation is well known as the Ambartsumian delay equation which has a particular application in Astrophysics. In this paper, the Laplace transform is successfully applied to solve the Pantograph delay equation. The solution is obtained in a closed series form in terms of exponential functions. This closed form reduces to the corresponding solution in the relevant literature for the Ambartsumian delay equation as a special case. In addition, the convergence of the obtained series is proved theoretically and validated graphically. Furthermore, the accuracy of the numerical results are estimated through several computations of the residual errors. It is shown that such residuals tend to zero, even in a huge domain. The obtained results reveal that the Laplace transform is a powerful approach to solve linear delay differential equations, including the Pantograph model.

The delay differential equations are of great importance in real-life phenomena. A special type of these equations is the Pantograph delay differential equation. Generally, solving a delay differential equation is a challenge, especially when the complexity of the delay terms increases. In this paper, the homotopy perturbation method is proposed to solve the Pantograph delay differential equation via two different canonical forms; thus, two types of closed-form solutions are determined. The first gives the standard power series solution while the second introduces the exponential function solution. It is declared that the current solution agrees with the corresponding ones in the literature in special cases. In addition, the properties of the solution are provided. Furthermore, the results are numerically validated through performing several comparisons with the available exact solutions. Moreover, the calculated residuals tend to zero, even in a huge domain, which reflects the high accuracy of the current analysis. The obtained results reveal the effectiveness and efficiency of the current analysis which can be further extended to other types of delay equations.

This paper considers a class of non-homogeneous fractional systems with harmonic terms by means of the Riemann–Liouville definition. Two different approaches are applied to obtain the dual solution of the studied class. The first approach uses the Laplace transform (LT) and the solution is given in terms of the Mittag-Leffler functions. The second approach avoids the LT and expresses the solution in terms of exponential and periodic functions which is analytic in the whole domain. The current methods determine the solution directly and efficiently. The results are applicable for other problems of higher order.

The main purpose of the present paper is to investigate the global asymptotic eigenvalue density of the fixed-trace generalized Gaussian ensemble of random matrices. To answer such a question, we begin with a complete study of the zeros of generalized Hermite orthogonal polynomials which are closely related to the eigenvalues in question.

This paper re-investigates the mathematical transport model of chlorine used as a water treatment model, when a variable order partial derivative is incorporated for describing the chlorine transport system. This model was introduced in the literature and governed by a fractional partial differential equation (FPDE) with prescribed boundary conditions. The obtained solution in the literature was based on implementing the Laplace transform (LT) combined with the method of residues and expressed in terms of regular exponential functions. However, the present analysis avoids such a method of residues, and thus a new analytical solution is introduced in this paper via Mittag-Leffler functions. Therefore, an effective approach is developed in this paper to solve the chlorine transport model with non-integer order derivative. In addition, our results are compared with several studies in the literature in case of integer-order derivative and the differences in results are explained.

In this paper, a new approach is developed to solve a class of first-order fractional initial value problems. The present class is of practical interest in engineering science. The results are based on the Riemann–Liouville fractional derivative. It is shown that the dual solution can be determined for the considered class. The first solution is obtained by means of the Laplace transform and expressed in terms of the Mittag–Leffler functions. The second solution was determined through a newly developed approach and given in terms of exponential and trigonometric functions. Moreover, the results reduce to the ordinary version as the fractional-order tends to unity. Characteristics of the dual solution are discussed in detail. Furthermore, the advantages of the second solution over the first one is declared. It is revealed that the second solution is real at certain values of the fractional-order. Such values are derived theoretically and accordingly, and the behavior of the real solution is shown through several plots. The present analysis may be introduced for obtaining the solution in a straightforward manner for the first time. The developed approach can be further extended to include higher-order fractional initial value problems of oscillatory types.