In this paper, the lognormal distribution is studied, and a new series representation is proposed. This series uses the powers of the bilinear function. From it, a simplified form is obtained and used to compute the Laplace transform of the distribut...
In this paper, we develop an analytical approximate solution for the nonlinear time-fractional Fisher’s equation using a right starting space function and a unique analytic-numeric technique referred to as the Laplace residual power series appr...
Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique for employing the Laplace transform LT in solving the non...
This article investigates different nonlinear systems of fractional partial differential equations analytically using an attractive modified method known as the Laplace residual power series technique. Based on a combination of the Laplace transforma...
The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to g...
In this paper, we present an efficient solution method for solving fractional system partial differential equations (FSPDEs) using the Laplace residual power series (LRPS) method. The LRPS method is a powerful technique for solving FSPDEs, as it allo...
In the present study, the exact solutions of the fractional three-dimensional (3D) Helmholtz equation (FHE) are obtained using the Laplace residual power series method (LRPSM). The fractional derivative is calculated using the Caputo operator. First,...
In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms o...
16 November 2025
This work introduces the General Residual Power Series Method (GRPSM) as a unified analytical framework encompassing the conventional Residual Power Series Method (RPSM) and its Laplace-like transform variants. By deriving a universal coefficient for...
Most physical phenomena are formulated in the form of non-linear fractional partial differential equations to better understand the complexity of these phenomena. This article introduces a recent attractive analytic-numeric approach to investigate th...
In this article, a hybrid numerical technique combining the Laplace transform and residual power series method is used to construct a series solution of the nonlinear fractional Riccati differential equation in the sense of Caputo fractional derivati...
In this paper, a system of coupled fractional neutron diffusion equations with delayed neutrons was solved efficiently by using a combination of residual power series and Laplace transform techniques, and the anomalous diffusion was considered by tak...
In this work, we first develop the modified time Caputo fractional Kawahara Equations (MTCFKEs) in the usual Hilbert spaces and extend them to analogous structures within the theory of Hilbert algebras. Next, we employ the residual power series metho...
This research focuses on finding multiple solutions (MSs) to nonlinear fractional boundary value problems (BVPs) through a new development, namely the predictor Laplace fractional power series method. This method predicts the missing initial values b...
In this paper, I utilize the Laplace residual power series method (LRPSM) along with a novel iteration technique to investigate the Fitzhugh-Nagumo equation within the framework of the Caputo operator. The Fitzhugh-Nagumo equation is a fundamental mo...
In this paper, we present the series solutions of the nonlinear time-fractional coupled Boussinesq-Burger equations (T-FCB-BEs) using Laplace-residual power series (L-RPS) technique in the sense of Caputo fractional derivative (C-FD). To assert the e...
In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive th...
Fractional-order differential equations are prevalent in many scientific fields; hence, their study has seen a renaissance in recent years. The fascinating realm of fractional calculus is explored in this research study, with particular emphasis on t...
For a regularly converging-in-C series A(z)=∑n=1∞anf(λnz), where f is an entire transcendental function, the asymptotic behavior of the function Mf−1(MA(r)), where Mf(r)=max{|f(z)|:|z|=r}, is investigated. It is proven that, under certain conditions...
In this article, a new higher-order convergence Laplace–Fourier method is developed to obtain the solutions of linear neutral delay differential equations. The proposed method provides more accurate solutions than the ones provided by the pure...
New estimates for the convergence abscissas of the multiple Laplace–Stieltjes integral are obtained. There is described the relationship between the integrand function, the Lebesgue–Stieltjes measure, and the abscissa of convergence of th...
The infinite series solution to the boundary-value problems of Laplace’s equation with discontinuous Dirichlet boundary conditions was found by using the basic method of separation of variables. The merit of this paper is that the closed-form s...
In the current analysis, a specific efficient and applicable novel solution approach, based on a fractional power series technique and Laplace transform operator, is considered to predict certain accurate approximate solutions (ASs) for a time-fracti...
In this paper, fractional-order system gas dynamics equations are solved analytically using an appealing novel method known as the Laplace residual power series technique, which is based on the coupling of the residual power series approach with the...
The objective of this work is to investigate analytical solutions of some models of cancer tumors using the Laplace residual power series method (LRPSM). The proposed method was effective and required simple calculations to find the analytic series s...
In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theor...
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. I...
In this research, we employ a dual-approach that combines the Laplace residual power series method and the novel iteration method in conjunction with the Caputo operator. Our primary objective is to address the solution of two distinct, yet intricate...
In this article, we present a modified strategy that combines the residual power series method with the Laplace transformation and a novel iterative technique for generating a series solution to the fractional nonlinear Belousov–Zhabotinsky (BZ...
In this research, the concept of nonlinear transfer function with nonlinear characteristics is introduced through the multidimensional Laplace transform and modal series (MS) method. The method of modal series is applied to the power systems dynamics...
The development of numeric-analytic solutions and the construction of fractional-order mathematical models for practical issues are of the greatest importance in a variety of applied mathematics, physics, and engineering problems. The Laplace residua...
In this study, we applied the Laplace residual power series method (LRPSM) to expand the solution of the nonlinear time-fractional coupled Hirota–Satsuma and KdV equations in the form of a rapidly convergent series while considering Caputo frac...
Building upon the previous research that solved neutron diffusion equations in simplified slab geometry, this study advances the field by addressing the more complex cylindrical geometry, focusing on neutron diffusion equations that are coupled with...
This article presents an idea of a new approach for the solitary wave solution of the modified Degasperis–Procesi (mDP) and modified Camassa–Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (LT) and homoto...
In this paper, an anomalous advection-dispersion model involving a new general Liouville–Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained wi...
This work provides exact and analytical approximate solutions for a non-linear time-fractional generalized biology population model (FGBPM) with suitable initial data under the time-Caputo fractional derivative, in view of a novel effective and appli...
In this work, we employed the Laplace transform of right-sided distributions in conjunction with the power series method to obtain distributional solutions to the modified Bessel equation and its related equation, whose coefficients contain the param...
An analytical method for solving the fractional diffusion–wave equation with a reaction is investigated. This approach is based on the Laplace transform and fractional series method. An analytical derivation for the proposed method is presented...
This work presents a reliable algorithm to obtain approximate analytical solutions for a strongly coupled system of singularly perturbed convection–diffusion problems, which exhibit a boundary layer at one end. The proposed method involves cons...
In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous metho...
The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. M...
The development of numeric-analytic solutions and the construction of fractional order mathematical models for practical issues are of the highest concern in a variety of physics, applied mathematics, and engineering applications. The nonlinear Kerst...
In this paper, we propose two efficient methods for solving the fractional-order Schrödinger–KdV system. The first method is the Laplace residual power series method (LRPSM), which involves expressing the solution as a power series and usi...
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...
The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y ( t ) + λ Γ ( α ) ∫ 0 t ( t − τ ) α − 1 y ( τ ) d τ = f ( t ) , ( t > 0 ) and its variants by fractional calculus. A...
This article employs the Laplace residual power series approach to study nonlinear systems of time-fractional partial differential equations with time-fractional Caputo derivative. The proposed technique is based on a new fractional expansion of the...
In this paper, we investigate the fractional-order Klein–Fock–Gordon equations on quantum dynamics using a new iterative method and residual power series method based on the Caputo operator. The fractional-order Klein–Fock–Gor...
Continuous real-time location data is very important in the big data era, but the privacy issues involved is also a considerable topic. It is not only necessary to protect the location privacy at each release moment, but also have to consider the imp...
Accurate forecasting of high-resolution particulate matter 2.5 (PM2.5) levels is essential for the development of public health policy. However, datasets used for this purpose often contain missing observations. This study presents a two-stage approa...
Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel functi...
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