Partial Differential Equations in Applied Mathematics最新文献

英文中文

Presentation of the efficient scheme for solving fractional order telegraph problems 介绍解决分数阶电报问题的高效方案

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-11-05 DOI: 10.1016/j.padiff.2024.100976

Wasim Sajjad Hussain , Sajjad Ali , Nahid Fatima , Kamal Shah , Thabet Abdeljawad

The Asymptotic Homotopy Perturbation Transform Method AHPTM is presented in this work. It is combined version of the Asymptotic Homotopy Perturbation Method AHPM and Laplace transformation. The focus of the work is the introduction of a new fast convergent scheme to obtain the solution of the fractional partial differential equations. Therefore, the first demonstration of the AHPTM is present for the solution of space-fractional telegraph equation (SFTE) in this work. The Caputo version of fractional derivatives has been utilized. Three test problems of the important fractional telegraph model were solved by this proposed scheme. The scheme of AHPTM worked without exploiting Ji. Huan He polynomials or Adomian polynomials. This application was elaborated by providing error estimates, a graphical presentation and tabulation of the results obtained by AHPTM. The comparison of results obtained by AHPTM with exact results is provided which indicated the accuracy of the scheme.

本研究提出了渐近同调扰动变换方法 AHPTM。它是渐近同调扰动法 AHPM 和拉普拉斯变换的组合版本。这项工作的重点是引入一种新的快速收敛方案，以获得分数偏微分方程的解。因此，在这项工作中，AHPTM 首次展示了空间-分数电报方程（SFTE）的解法。本文采用了分数导数的 Caputo 版本。该方案解决了重要分数电报模型的三个测试问题。AHPTM 方案没有利用 Ji.Huan He 多项式或 Adomian 多项式。通过对 AHPTM 得出的结果进行误差估计、图形展示和列表，对这一应用进行了详细阐述。AHPTM 得出的结果与精确结果进行了比较，表明了该方案的准确性。

引用次数: 0

Analysis of Cauchy reaction-diffusion equations involving Atangana-Baleanu fractional operator 涉及阿坦加纳-巴列阿努分数算子的考奇反应-扩散方程分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-11-03 DOI: 10.1016/j.padiff.2024.100981

Hassan Kamil Jassim, Ali Latif Arif

This study investigates the Cauchy reaction-diffusion equation (CRDE) with the Atangana-Baleanu differential operator. The existence and uniqueness of solutions to fractional starting value issues are begun using the fixed-point theorem and contraction principle, respectively. The proposed study uses the natural variation iteration technique (NVIM) to get an approximate solution for nonlinear fractional reaction-diffusion equations. This study's approximate answers are compared to other solutions found using known methodologies, and the results are discussed. The devised technique has benefits in terms of accuracy and computational cost efficiency, which may be used to solve nonlinear fractional reaction-diffusion equations.

本研究探讨了带有 Atangana-Baleanu 微分算子的 Cauchy 反应扩散方程（CRDE）。分别利用定点定理和收缩原理开始研究分数起始值问题解的存在性和唯一性。建议的研究使用自然变异迭代技术（NVIM）来获得非线性分数反应扩散方程的近似解。本研究的近似解与使用已知方法找到的其他解进行了比较，并对结果进行了讨论。所设计的技术在精度和计算成本效率方面都有优势，可用于求解非线性分数反应扩散方程。

引用次数: 0

Simulations for the Schrödinger–Hirota equation arising in nonlinear optics in the presence of chromatic dispersion 存在色散的非线性光学中产生的薛定谔-希罗塔方程的模拟

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-11-01 DOI: 10.1016/j.padiff.2024.100969

Maasoomah Sadaf , Saima Arshed , Ghazala Akram , Muhammad Abdaal Bin Iqbal , Hijaz Ahmad , Mohamed R. Ali

The main objective of this work is to study the accurate traveling wave behavior of the optical pulses described by the Schrödinger–Hirota equation taking into account the chromatic dispersion term. This study uses the extended-

(\frac{G^{'}}{G^{2}})

and the

exp (- ϕ (ϖ))

-expansion methods to get the exact closed form wave solutions to the Schrödinger–Hirota problem. Nonlinearity with Kerr rule is used to analyze the aforementioned model, leading to some novel conclusions. A variety of dynamical wave patterns have been observed through graphical simulations of the retrieved solutions. The reported results may be helpful in further explanation in optical fibers, communication systems and nonlinear optics.

这项工作的主要目的是研究薛定谔-希罗塔方程描述的光脉冲的精确行波行为，同时考虑色度色散项。本研究使用扩展 G′G2 和 exp(-j(ϖ)) 展开方法来获得薛定谔-Hirota 问题的精确闭式波解。利用克尔规则的非线性分析了上述模型，并得出了一些新的结论。通过对检索到的解进行图形模拟，观察到了各种动态波形。报告的结果可能有助于进一步解释光纤、通信系统和非线性光学。

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引用次数: 0

Integrability conditions for Boussinesq type systems Boussinesq 型系统的可积分性条件

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-29 DOI: 10.1016/j.padiff.2024.100959

R. Hernández Heredero , V. Sokolov

The symmetry approach to the classification of evolution integrable partial differential equations (see, for example (Mikhailov et al.,1991)) produces an infinite series of functions, defined in terms of the right hand side, that are conserved densities of any equation having infinitely many infinitesimal symmetries. For instance, the function

\frac{\partial f}{\partial u_{x}}

has to be a conserved density of any integrable equation of the KdV type

u_{t} = u_{x x x} + f (u, u_{x})

. This fact imposes very strong conditions on the form of the function

f

. In this paper we construct similar canonical densities for equations of the Boussinesq type. In order to do that, we write the equations as evolution systems and generalise the formal diagonalisation procedure proposed in Mikhailov et al. (1987) to these systems.

对可演化积分偏微分方程进行分类的对称性方法（例如，见 Mikhailov 等人，1991 年）产生了一个无穷系列的函数，这些函数定义在右边，是具有无限多无穷小对称性的任何方程的守恒密度。例如，函数 ∂f∂ux 必须是任何 KdV 型可积分方程 ut=uxxx+f(u,ux) 的守恒密度。这一事实对函数 f 的形式提出了非常苛刻的条件。在本文中，我们将为布西内斯克方程构建类似的典型密度。为此，我们将方程写成演化系统，并将 Mikhailov 等人（1987 年）提出的正式对角化程序推广到这些系统中。

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引用次数: 0

Effects of exponentially stretching sheet for MHD Williamson nanofluid with chemical reaction and thermal radiative 带有化学反应和热辐射的 MHD Williamson 纳米流体的指数拉伸薄片效应

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-29 DOI: 10.1016/j.padiff.2024.100975

S P Pallavi , M.B Veena , Jagadish. V. Tawade , Nitiraj Kulkarni , Sami Ullah Khan , M. Waqas , Manish Gupta , Saja Abdulrahman Althobaiti

This paper explores the combined effects of heat radiation, viscous dissipation, and chemical reactions on the steady flow of Williamson nanofluid over an exponentially stretched sheet. The Governing non-linear Partial Differential Equations (PDE's), converted to couple nonlinear Ordinary ODE's by using similarity transformation, which are solved numerically using the Runge-Kutta-Fehlberg method along with the shooting technique. The study shows detailed analysis of the behaviour of Williamson nanofluid under the influence of thermal radiation and magnetic fields, having relevant industrial applications in cooling technologies and polymer processing. The results show that increasing the magnetic field parameter reduces the fluid velocity, while higher thermal radiation and Brownian motion parameters significantly enhance heat transfer rate withing the boundary region.

本文探讨了热辐射、粘性耗散和化学反应对威廉姆森纳米流体在指数拉伸片上稳定流动的综合影响。通过相似性转换，将支配性非线性偏微分方程（PDE）转换为耦合非线性普通 ODE，并使用 Runge-Kutta-Fehlberg 方法和射击技术对其进行数值求解。研究详细分析了威廉姆森纳米流体在热辐射和磁场影响下的行为，这些行为在冷却技术和聚合物加工中具有相关的工业应用。结果表明，增加磁场参数会降低流体速度，而增加热辐射和布朗运动参数则会显著提高边界区域的传热速率。

引用次数: 0

Stochastic stability and global dynamics of a mathematical model for drug use: Statistical sensitivity analysis via PRCC 药物使用数学模型的随机稳定性和全局动态性：通过 PRCC 进行统计敏感性分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-28 DOI: 10.1016/j.padiff.2024.100964

Sara Soulaimani , Abdelilah Kaddar , Fathalla A. Rihan

This article examines the stochastic stability and global dynamics of a mathematical model of drug use. The model divides the population into five compartments current drug users, temporarily abstinent drug users, permanently abstinent drug users, and drug users in rehabilitation. Using Brownian motion, deterministic equations are extended to incorporate stochastic perturbations, capturing real-life uncertainties in drug use within compartments. An analysis of Lyapunov functions is used to determine the global stability of the model. By introducing stochastic elements into the model, we can examine its stability under random perturbations. A global sensitivity analysis, including PRCC results, is conducted to confirm the robustness of the model. Stable drug-free and drug-present equilibria can be maintained in both deterministic and stochastic environments. Numerical simulations illustrate the impact of various parameters on population dynamics and rehabilitation program effectiveness.

本文研究了一个吸毒数学模型的随机稳定性和全局动力学。该模型将人口分为五个部分：当前吸毒者、暂时戒毒者、永久戒毒者和康复中的吸毒者。利用布朗运动，将确定性方程扩展到随机扰动，从而捕捉到现实生活中各分区内毒品使用的不确定性。对 Lyapunov 函数的分析用于确定模型的全局稳定性。通过在模型中引入随机因素，我们可以检验模型在随机扰动下的稳定性。我们还进行了包括 PRCC 结果在内的全局敏感性分析，以确认模型的稳健性。在确定性和随机环境中，都能保持稳定的无药平衡和有药平衡。数值模拟说明了各种参数对种群动态和康复计划有效性的影响。

引用次数: 0

Application of fractional differential transform method and Bell polynomial for solving system of fractional delay differential equations 应用分数微分变换法和贝尔多项式求解分数延迟微分方程系

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-28 DOI: 10.1016/j.padiff.2024.100971

Sandeep Kumar Yadav, Giriraj Methi

In this article, a new numerical technique is presented to obtain numerical solution of a system of fractional delay differential equations (FDDE’s) involving proportional and time dependent delay terms. The fractional derivative is used in Caputo sense. The proposed technique is the combination of fractional differential transform and Bell polynomial. The existence and uniqueness results are discussed for FDDE’s. Three numerical problems are discussed to show reliability and efficiency of the method. Numerical results are compared with exact and Matlab DDENSD solution. The main advantage of the present method is handing effectively the nonlinear terms present in the FDDEs by using Bell polynomial. The present method can deal with both linear and nonlinear FDDEs. The convergence result is discussed, and error analysis is presented in detail.

本文介绍了一种新的数值技术，用于获得涉及比例和时间相关延迟项的分数延迟微分方程（FDDE）系统的数值解。分数导数是在 Caputo 意义上使用的。所提出的技术是分数微分变换和贝尔多项式的结合。讨论了 FDDE 的存在性和唯一性结果。讨论了三个数值问题，以显示该方法的可靠性和效率。数值结果与精确解法和 Matlab DDENSD 解法进行了比较。本方法的主要优点是利用贝尔多项式有效地处理了 FDDE 中存在的非线性项。本方法既能处理线性 FDDE，也能处理非线性 FDDE。本文讨论了收敛结果，并详细介绍了误差分析。

引用次数: 0

Noncommutative solutions to the local tetrahedron equation 局部四面体方程的非交换解

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-28 DOI: 10.1016/j.padiff.2024.100968

M. Chirkov

We study the solutions of the local Zamolodchikov tetrahedron equation on noncommutative groups and division rings in the form of correspondences derived from 3 × 3 matrices with free noncommutative variables. The complete set of generators for 4-simplex maps that adhere to the local tetrahedron equation is presented. We study the difference in classification between commutative and noncommutative cases. Additionally, we introduce a procedure for obtaining novel 4-simplex maps associated with known tetrahedron maps. Also, we introduce the “conditional

n

-simplex maps” and study the case of 4-simplex maps via examples. Lastly, several new 4-simplex maps on noncommutative groups are constructed.

我们研究了非交换群和划分环上局部扎莫洛奇科夫四面体方程的解，其形式是由具有自由非交换变量的 3 × 3 矩阵派生的对应关系。我们提出了符合局部四面体方程的 4-复数映射的全套生成器。我们研究了交换和非交换情况下的分类差异。此外，我们还介绍了一种获得与已知四面体映射相关的新型 4-simplex映射的程序。此外，我们还介绍了 "条件 n-复数映射"，并通过实例研究了 4-复数映射的情况。最后，我们还构建了非交换群上的几个新的 4-simplex 映射。

引用次数: 0

Asymptotic behavior of interior peaked solutions for a slightly subcritical Neumann problem 轻微次临界新曼问题内部峰值解的渐近行为

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.padiff.2024.100920

Fatimetou Mohamed Salem

In this paper, we study the asymptotic behavior of solutions of the Neumann problem

(P_{ɛ})

- Δ u + V (x) u = u^{p - ɛ}

u > 0

Ω

\partial u / \partial ν = 0

\partial Ω

, where

Ω

is a smooth bounded domain in

R^{n}

n \geq 6

p + 1 = 2 n / (n - 2)

is the critical Sobolev exponent,

ɛ

is a small positive real and

V

is a smooth positive function defined on

\bar{Ω}

. We give a precise location of interior blow up points and blow up rates when the number of concentration points is less than or equal to 2. The proof strategy is based on a refined blow up analysis in the neighborhood of bubbles.

本文研究 Neumann 问题 (Pɛ) 解的渐近行为：-Δu+V(x)u=up-ɛ, u>0 in Ω, ∂u/∂ν=0 on ∂Ω，其中 Ω 是 Rn 中的光滑有界域，n≥6，p+1=2n/(n-2) 是临界 Sobolev 指数，ɛ 是小正实数，V 是定义在 Ω¯ 上的光滑正函数。我们给出了当集中点个数小于或等于 2 时内部炸裂点的精确位置和炸裂率，证明策略基于气泡邻域的精细炸裂分析。

引用次数: 0

Fractal-view and convergence of fractional order cauchy reaction-diffusion equations using semi-analytical technique 利用半解析技术研究分数阶考希反应扩散方程的分形视图和收敛性

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.padiff.2024.100974

H.M. Younas , Kousar Yousaf , Imran Siddique , Shaukat Iqbal

This article solves time-fractional order Cauchy reaction-diffusion equations in approximation using the Optimal Homotopy Asymptotic Method (OHAM).The exact solutions and approximate third-order results achieved using OHAM are compared. It has been noted that for partial time order Cauchy equations for reaction-diffusion, OHAM findings exhibit a substantial convergence rate. Plotting the outcomes of the solutions and tabulating the relative errors are done. In order to find the mentioned solutions Mathematica has been used.

本文利用最优同调渐近法（OHAM）近似求解时分数阶 Cauchy 反应扩散方程，并比较了利用 OHAM 获得的精确解和近似三阶结果。我们注意到，对于部分时序的考希反应扩散方程，OHAM 的研究结果显示出很高的收敛速度。我们绘制了求解结果图，并将相对误差列表。为了找到上述解决方案，我们使用了 Mathematica。

引用次数: 0

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