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This paper considers the use of the Laplace residual power series method to solve linear Volterra integrodifferential equations. It demonstrates how well the approach holds up to both linear and nonlinear fractional starting value situations. Important contributions are comparing the Laplace residual power series approach with other numerical methods and providing evidence of its reliability through recent investigations. [ABSTRACT FROM AUTHOR]

This article presents the modified residual power series approach using Laplace transform, the method is used to solve partial integro differential equations. The basic definitions and theorems related to the method are presented and discussed. Moreover, the steps of the method are utilized and applied to solve various examples. [ABSTRACT FROM AUTHOR]

In this study, we present a modified analytical approximation method to find the time-fractional Sharma-Tasso-Olever issue solving. In order to tackle nonlinear fractional differential equations that arise in a variety of physical processes, we begin by providing an alternate foundation for the Laplace Residual Power Series Technique (LRPSM). Thus, the generalized Taylor series equation and residual functions serve as the foundation for this approach. More precisely, our approach and the suggested solution produce good results. Moreover, the reliability, effectiveness, and simplicity of this approach are demonstrated for all classes of fractional nonlinear issues that arise in technological and scientific fields. Two examples are provided to exemplify how the considered scheme works in calculating various types of fractional ordinary differential equations. Finally, the obtained results in this article are compared with other methods such as Residual Power Series (RPS), Variational Iteration Method (VIM), and Homotopy Perpetration Method (HPM). The consequences of our method are good and effective. [ABSTRACT FROM AUTHOR]

In this research article, we present a novel Non-Linear Fractional Order Tuberculosis mathematical model (NLFOTB) and introduce an efficient technique to obtain its solution. Fractional Order Models (FOMs) have garnered significant attention in contemporary research due to their widespread applicability. We address the challenge of solving the coupled Initial Value Problems (IVPs) associated with NLFOTB models by utilizing the groundbreaking LRPS method, which combines the RPS approach with the Laplace transform operator. This innovative approach generates approximate solutions in rapidly converging series forms, offering enhanced efficiency and reduced computational effort compared to conventional methods. Through the implementation of the LRPS method, we successfully derive an approximate solution for the NLFOTB model, contributing significantly to the field. Furthermore, our proposed approach demonstrates its efficacy in accurately capturing the dynamics of Tuberculosis (TB) through extensive computations and graphical representations, contributing to a deeper understanding of TB dynamics within a mathematical framework. Additionally, the LRPS method shows promise in tackling real world problems involving differential equations of various orders. Future investigations can extend the application of the LRPS method to explore other Fractional Order Models, further validating its effectiveness in a wide range of epidemic scenarios. Consequently, our study not only provides valuable insights into Tuberculosis dynamics but also introduces a powerful computational tool applicable to various practical problems in diverse disciplines, making a substantial contribution to the field of mathematical modeling and computation. [ABSTRACT FROM AUTHOR]

In this paper, a biophysical fractional diffusive cancer model with virotherapy is thoroughly analyzed and analytically simulated. The goal of this biophysical model is to represent both the dynamics of cancer development and the results of virotherapy, which uses viruses to target and destroy cancer cells. The Caputo sense is applied to the fractional derivatives. We look at the governing model's existence and uniqueness. For analytical solutions, the Laplace residual power series approach is used. The study investigates the model's dynamic behavior, shedding light on the development of cancer and the effects of virotherapy. The research advances our knowledge of cancer modeling and treatment options. Numerical simulations show the agreement between the analytical results and the related numerical solutions, proving the usefulness of the analytical solution.

In this paper, a biophysical fractional diffusive cancer model with virotherapy is thoroughly analyzed and analytically simulated. The goal of this biophysical model is to represent both the dynamics of cancer development and the results of virotherapy, which uses viruses to target and destroy cancer cells. The Caputo sense is applied to the fractional derivatives. We look at the governing model's existence and uniqueness. For analytical solutions, the Laplace residual power series approach is used. The study investigates the model's dynamic behavior, shedding light on the development of cancer and the effects of virotherapy. The research advances our knowledge of cancer modeling and treatment options. Numerical simulations show the agreement between the analytical results and the related numerical solutions, proving the usefulness of the analytical solution. [ABSTRACT FROM AUTHOR]

This current work presents a comparative study of the fractional-order Cahn-Allen (CA) equation, where the non-integer derivative is taken in the Caputo sense.The Cahn-Allen equation is an equation that assists in the comprehension of phase transitions and pattern formation in physical systems. This equation describes how different phases of matter, such as solids and liquids, change and interact throughout time. We employ two analytical methods: the Laplace Residual Power Series Method (LRPSM) and the New Iterative Method (NIM), to solve the proposed model. The LRPSM is a combination of the Laplace Transform and the Residual Power Series Method, while the New Iterative Method is a modified form of the Adomian Decomposition Method that does not require any type of polynomial or digitization. For the purpose of accuracy and reliability, we compare our findings with other methods and the exact solution used in the literature. Additionally, 2D and 3D plots are generated for various fractional order values denoted as p. These plots illustrate that as the fractional order p approaches 1, the graph of the approximate solution gradually coincides with the graph of the exact solution.

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 generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach's reliability, capability, and efficiency. [ABSTRACT FROM AUTHOR]

Constructing mathematical models of fractional order for real-world problems and developing numeric-analytic solutions are extremely significant subjects in diverse fields of physics, applied mathematics and engineering problems. In this work, a novel analytical treatment technique called the Laplace residual power series (LRPS) technique is performed to produce approximate solutions for a non-linear time-fractional gas dynamics equation (FGDE) in a multiple fractional power series (MFPS) formula. The LRPS technique is a coupling of the RPS approach with the Laplace transform operator. The implementation of the proposed technique to handle time-FGDE models is introduced in detail. The MFPS solution for the target model is produced by solving it in the Laplace space by utilizing the limit concept with fewer computations and more accuracy. The applicability and performance of the technique have been validated via testing three attractive initial value problems for non-linear FGDEs. The impact of the fractional order β on the behavior of the MFPS approximate solutions is numerically and graphically described. The jth MFPS approximate solutions were found to be in full harmony with the exact solutions. The solutions obtained by the LRPS technique indicate and emphasize that the technique is easy to perform with computational efficiency for different kinds of time-fractional models in physical phenomena.

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 taking the non-Gaussian case with different values of fractional parameter α. The Laplace residual power series method (LRPSM) does not require differentiation, conversion, or discretization for the assumed conditions, so the approach is simple and suitable for solving higher-order fractional differential equations. To assure the theoretical results, two different neutron flux initial conditions were presented numerically, where the needed Mathematica codes were performed using essential nuclear reactor cross-section data, and the results for different values of times were tabulated and graphically figured out. Finally, it must be noted that the results align with the Adomian decomposition method. [ABSTRACT FROM AUTHOR]

In this study, different systems of linear and non-linear fractional initial value problems are solved analytically utilizing an attractive novel technique so-called the Laplace residual power series approach, and which is based on the coupling of the residual power series approach with the Laplace transform operator to generate analytical and approximate solutions in fast convergent series forms by using the concept of the limit with less time and effort compared with the residual power series technique. To confirm the simplicity, performance, and viability of the proposed technique, three problems are tested and simulated. Analysis of the obtained results reveals that the aforesaid technique is straightforward, accurate, and suitable to investigate the solutions of the non-linear physical and engineering problems.

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 generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach’s reliability, capability, and efficiency.

In this study, different systems of linear and non-linear fractional initial value problems are solved analytically utilizing an attractive novel technique so-called the Laplace residual power series approach, and which is based on the coupling of the residual power series approach with the Laplace transform operator to generate analytical and approximate solutions in fast convergent series forms by using the concept of the limit with less time and effort compared with the residual power series technique. To confirm the simplicity, performance, and viability of the proposed technique, three problems are tested and simulated. Analysis of the obtained results reveals that the aforesaid technique is straightforward, accurate, and suitable to investigate the solutions of the non-linear physical and engineering problems. [ABSTRACT FROM AUTHOR]

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 taking the non-Gaussian case with different values of fractional parameter α. The Laplace residual power series method (LRPSM) does not require differentiation, conversion, or discretization for the assumed conditions, so the approach is simple and suitable for solving higher-order fractional differential equations. To assure the theoretical results, two different neutron flux initial conditions were presented numerically, where the needed Mathematica codes were performed using essential nuclear reactor cross-section data, and the results for different values of times were tabulated and graphically figured out. Finally, it must be noted that the results align with the Adomian decomposition method.

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 Maclurian series, which provides a rapid convergence series solution where the coefficients of the proposed fractional expansion are computed with the limit concept. The nonlinear systems studied in this work are the Broer-Kaup system, the Burgers’ system of two variables, and the Burgers’ system of three variables, which are used in modeling various nonlinear physical applications such as shock waves, processes of the wave, transportation of vorticity, dispersion in porous media, and hydrodynamic turbulence. The results obtained are reliable, efficient, and accurate with minimal computations. The proposed technique is analyzed by applying it to three attractive problems where the approximate analytical solutions are formulated in rapid convergent fractional Maclurian formulas. The results are studied numerically and graphically to show the performance and validity of the technique, as well as the fractional order impact on the behavior of the solutions. Moreover, numerical comparisons are made with other well-known methods, proving that the results obtained in the proposed technique are much better and the most accurate. Finally, the obtained outcomes and simulation data show that the present method provides a sound methodology and suitable tool for solving such nonlinear systems of time-fractional partial differential equations.