Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

Chen, Bochao; Qin, Li; Xu, Fei; Zu, Jian

doi:10.1155/2018/2394735

Cited by 11 publications

(5 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is worth noting that the integer-order solution (73) coincides with the result proposed by [48,49].…”

Section: Test Examplessupporting

confidence: 80%

“…Hence, the MED terms of global TIC, MAD and EVAF physical quantities reside 10 −7 −10 −8 , 10 −9 −10 −10 , and 10 −9 −10 −10 , respectively, whilst the global S.I.R formula of TIC, MAD and EVAF reside 10 −5 −10 −6 , 10 −9 −10 −10 , and 10 −9 −10 −10 , respectively for Examples 1-5 dynamical framework. The correlation has proven that the indicated method and [29,[47][48][49] produce the same results, demonstrating the efficiency and dependability of the ARARPSM. The accuracy, exactness, and efficacy of the interconnected supercomputing algorithms of ARARPSMA for tackling the nonlinear wave-likeand heat-like systems are further indicated by standard measures such as G-TIC, G-MAD, and G-EVAF that are similar to their optimal setting.…”

Section: Exactmentioning

confidence: 66%

“…Comparison of statistical data analysis presentations of the time-fractional nonlinear (3 + 1) wave-like system in Example 4 that depend on the fractional-order and averages such as mean, mean deviation, semi-inter quartile range, and standard deviation with different values of t, u with v = w = 0.1 and the results proposed by[48,49] …”

mentioning

confidence: 98%

See 2 more Smart Citations

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

Chu,

Sultana,

Karim

et al. 2024

Computer Modeling in Engineering &Amp; Sciences

View full text Add to dashboard Cite

The goal of this research is to develop a new, simplified analytical method known as the ARA-residue power series method for obtaining exact-approximate solutions employing Caputo type fractional partial differential equations (PDEs) with variable coefficient. ARA-transform is a robust and highly flexible generalization that unifies several existing transforms. The key concept behind this method is to create approximate series outcomes by implementing the ARA-transform and Taylor's expansion. The process of finding approximations for dynamical fractional-order PDEs is challenging, but the ARA-residual power series technique magnifies this challenge by articulating the solution in a series pattern and then determining the series coefficients by employing the residual component and the limit at infinity concepts. This approach is effective and useful for solving a massive class of fractional-order PDEs. Five appealing implementations are taken into consideration to demonstrate the effectiveness of the projected technique in creating solitary series findings for the governing equations with variable coefficients. Additionally, several visualizations are drawn for different fractional-order values. Besides that, the estimated findings by the proposed technique are in close agreement with the exact outcomes. Finally, statistical analyses further validate the efficacy, dependability and steady interconnectivity of the suggested ARA-residue power series approach.

show abstract

“…It is worth noting that the integer-order solution (73) coincides with the result proposed by [48,49].…”

Section: Test Examplessupporting

confidence: 80%

Section: Exactmentioning

confidence: 66%

See 1 more Smart Citation

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

Chu,

Sultana,

Karim

et al. 2024

Computer Modeling in Engineering &Amp; Sciences

View full text Add to dashboard Cite

show abstract

“…Nevertheless, while solving fractional-order PDE, say in two variables, this method assumes that one of the independent variables has a representation in a fractional power series, and the second independent variable is handled as a coefficient variable, which is roughly derived from the variation in the given fractional-order PDE based on the initial or boundary condition. For example, see authors in [16][17][18][19]. The same method was used by [20] to solve nonlinear fractional-order PDEs.…”

Section: Introductionmentioning

confidence: 99%

The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection

Barnes

Anokye

Iddrisu³

et al. 2023

Advances in Mathematical Physics

View full text Add to dashboard Cite

This paper provides a mathematical fractional-order model that accounts for the mindset of patients in the transmission of COVID-19 disease, the continuous inflow of foreigners into the country, immunization of population subjects, and temporary loss of immunity by recovered individuals. The analytic solutions, which are given as series solutions, are derived using the fractional power series method (FPSM) and the residual power series method (RPSM). In comparison, the series solution for the number of susceptible members, using the FPSM, is proportional to the series solution, using the RPSM for the first two terms, with a proportional constant of ψ Γ n α + 1 , where ψ is the natural birth rate of the baby into the susceptible population, Γ is the gamma function, n is the n th term of the series, and α is the fractional order as the initial number of susceptible individuals approaches the population size of Ghana. However, the variation in the two series solutions of the number of members who are susceptible to the COVID-19 disease begins at the third term and continues through the remaining terms. This is brought on by the nonlinear function present in the equation for the susceptible subgroup. The similar finding is made in the series solution of the number of exposed individuals. The series solutions for the number of deviant people, the number of nondeviant people, the number of people quarantined, and the number of people recovered using the FPSM are unquestionably almost identical to the series solutions for same subgroups using the RPSM, with the exception that these series solutions have initial conditions of the subgroup of the population size. It is observed that, in this paper, the series solutions of the nonlinear system of fractional partial differential equations (PDEs) provided by the RPSM are more in line with the field data than the series solutions provided by the FPSM.

show abstract

“…The Adomian decomposition method (ADM) [20] and the variational iteration method [21,22] are also mentioned in many contexts. The residual power series method (RPSM) is one of those techniques which quite suits nonlinear fractional differential equations [23][24][25][26][27][28][29][30]. Generalized from the classical power series method, the solution is written on the form of fractional power series.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solution of Nonlinear Fractional Volterra Population Growth Model Using the Modified Residual Power Series Method

Dunnimit

Wiwatwanich

2020

Symmetry

View full text Add to dashboard Cite

In this paper, we introduce an analytical approximate solution of nonlinear fractional Volterra population growth model based on the Caputo fractional derivative and the Riemann fractional integral of the symmetry order. The residual power series method and Adomain decomposition method are implemented to find an approximate solution of this problem. The convergence analysis of the proposed technique has been proved. A numerical example is given to illustrate the method.

show abstract

Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

Cited by 11 publications

References 32 publications

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection

Analytical Solution of Nonlinear Fractional Volterra Population Growth Model Using the Modified Residual Power Series Method

Contact Info

Product

Resources

About