A new approach for solving multi-pantograph type delay differential equations

<p>In this paper, a modified procedure based on the residual power series method (RPSM) was implemented to achieve approximate solution with high degree of accuracy for a system of multi-pantograph type delay differential equations (DDEs). This modified procedure is considered as a hybrid technique used to improve the curacy of the standard RPSM by combining the RPSM, Laplace transform and Pade approximant to be a powerful technique that can be solve the problems directly without large computational work, also even enlarge domain and leads to very accurate solutions or gives the exact solutions which is consider the best advantage of this technique. Some numerical applications are illustrated and numerical results are provided to prove the validity and the ability of this technique for this type of important differential equation that appears in different applications in engineering and control system.</p>

An Improved Version of Residual Power Series Method for Space-Time Fractional Problems

The task of present research is to establish an enhanced version of residual power series (RPS) technique for the approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameter λ . The parameter λ allows us to establish the best numerical solutions for space-time fractional differential equations (STFDE). Since each problem has different Dirichlet boundary conditions, the best choice of the parameter λ depends on the problem. This is the major contribution of this research. The illustrated examples also show that the best approximate solutions of various problems are constructed for distinct values of parameter λ . Moreover, the efficiency and reliability of this technique are verified by the numerical examples.

A Novel Numerical Approach in Solving Fractional Neutral Pantograph Equations via the ARA Integral Transform

In this article, a new, attractive method is used to solve fractional neutral pantograph equations (FNPEs). The proposed method, the ARA-Residual Power Series Method (ARA-RPSM), is a combination of the ARA transform and the residual power series method and is implemented to construct series solutions for dispersive fractional differential equations. The convergence analysis of the new method is proven and shown theoretically. To validate the simplicity and applicability of this method, we introduce some examples. For measuring the accuracy of the method, we make a comparison with other methods, such as the Runge–Kutta, Chebyshev polynomial, and variational iterative methods. Finally, the numerical results are demonstrated graphically.

Numerical investigation of fractional model of Phytoplankton–Toxic Phytoplankton–Zooplankton system with convergence analysis

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time [Formula: see text] on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.

Review of potential flow solutions for velocity and shape of long isolated bubbles in vertical pipes

The motion of elongated gas bubbles in vertical pipes has been studied extensively over the past century. A number of empirical and numerical correlations have emerged out of this curiosity; amongst them, analytical solutions have been proposed. A review of the major results and resolution methods based on a potential flow theory approach is presented in this article. The governing equations of a single elongated gas bubble rising in a stagnant or moving liquid are given in the potential flow formalism. Two different resolution methods (the power series method and the total derivative method) are studied in detail. The results (velocity and shape) are investigated with respect to the surface tension effect. The use of a new multi-objective solver coupled with the total derivative method improves the research of solutions and demonstrates its validity for determining the bubble velocity. This review aims to highlight the power of analytical tools, resolution methods and their associated limitations behind often well-known and wide-spread results in the literature.

Approximate and Exact Solutions to Fractional Order Cauchy Reaction-Diffusion Equations by New Combine Techniques

In this paper, we present a simple and efficient novel semianalytic method to acquire approximate and exact solutions for the fractional order Cauchy reaction-diffusion equations (CRDEs). The fractional order derivative operator is measured in the Caputo sense. This novel method is based on the combinations of Elzaki transform method (ETM) and residual power series method (RPSM). The proposed method is called Elzaki residual power series method (ERPSM). The proposed method is based on the new form of fractional Taylor’s series, which constructs solution in the form of a convergent series. As in the RPSM, during establishing the coefficients for a series, it is required to compute the fractional derivatives every time. While ERPSM only requires the concept of the limit at zero in establishing the coefficients for the series, consequently scarce calculations give us the coefficients. The recommended method resolves nonlinear problems deprived of utilizing Adomian polynomials or He’s polynomials which is the advantage of this method over Adomain decomposition method (ADM) and homotopy-perturbation method (HTM). To study the effectiveness and reliability of ERPSM for partial differential equations (PDEs), absolute errors of three problems are inspected. In addition, numerical and graphical consequences are also recognized at diverse values of fractional order derivatives. Outcomes demonstrate that our novel method is simple, precise, applicable, and effectual.

An Accurate Estimation of the Eigenfrequency of an Offshore Wind Turbine Considered as the Stepped Euler-Bernoulli Beam in Three-Spring Flexible Foundation Using the Power Series Method

The present study employs the power series method (PSM) to accurately predict the natural frequencies of eleven offshore wind turbines (OWT). This prediction is very important as it helps in the quick verification of experimental or finite element results. This study idealizes the OWT as a stepped Euler-Bernoulli beam carrying a top mass and connected at its bottom to a flexible foundation. The first part of the beam represents a monopile and the transition piece while its second part is a tower. The foundation is modeled using three springs (lateral, rotational, and cross-coupling springs). This work’s aim is at improving therefore the previous researches, in which the whole wind turbine was taken as a single beam, with a tower being tapered and its wall thickness being negligible compared to its diameter. In order to be closer to real-life OWT, three profiles of the tapered tower are explored: case 1 considers a tower with constant thickness along its height. Case 2 assumes a tower’s thickness being negligible compared to its mean diameter, while case 3 describes the tower as a tapered beam with varying thickness along its height. Next, the calculated natural frequencies are compared to those obtained from measurements. Results reveal that case 2, used by previous researches, was only accurate for OWT with tower wall thickness lower than 15 mm. Frequencies produced in case 3 are the most accurate as the relative error is up to 0.01%, especially for the OWT with thicknesses higher or equal to 15 mm. This case appears to be more realistic as, practically, wall thickness of a wind tower varies with its height. The tower-to-pile thickness ratio is an important design parameter as it highly has impact on the natural frequency of OWT, and must therefore be taken into account during the design as well as lateral and rotational coupling springs.

Solution of Fractional Partial Differential Equations Using Fractional Power Series Method

In this paper, we are presenting our work where the noninteger order partial differential equation is studied analytically and numerically using the noninteger power series technique, proposed to solve a noninteger differential equation. We are familiar with a coupled system of the nonlinear partial differential equation (NLPDE). Noninteger derivatives are considered in the Caputo operator. The fractional-order power series technique for finding the nonlinear fractional-order partial differential equation is found to be relatively simple in implementation with an application of the direct power series method. We obtained the solution of nonlinear dispersive equations which are used in electromagnetic and optics signal transformation. The proposed approach of using the noninteger power series technique appears to have a good chance of lowering the computational cost of solving such problems significantly. How to paradigm an initial representation plays an important role in the subsequent process, and a few examples are provided to clarify the initial solution collection.