Order Partial Differential Equations Research Articles

Laplace decomposition for solving nonlinear system of fractional order partial differential equations

In the present article a modified decomposition method is implemented to solve systems of partial differential equations of fractional-order derivatives. The derivatives of fractional-order are expressed in terms of Caputo operator. The validity of the proposed method is analyzed through illustrative examples. The solution graphs have shown a close contact between the exact and LADM solutions. It is observed that the solutions of fractional-order problems converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested technique. Due to better accuracy and straightforward implementation, the extension of the present method can be made to solve other fractional-order problems.

Isogeometric analysis of insoluble surfactant spreading on a thin film

In this paper we tackle the problem of surfactant spreading on a thin liquid film in the framework of isogeometric analysis. We consider a mathematical model that describes this phenomenon as an initial boundary value problem (IBVP) that includes two coupled fourth order partial differential equations (PDEs), one for the film height and one for the surfactant concentration. In order to solve this problem numerically, it is customary to transform it into a mixed problem that includes at most second order PDEs. However, the higher-order continuity of the approximation functions in Isogeometric Analysis (IGA) allows us to deal with the weak form of the fourth order PDEs directly, without the need of resorting to mixed methods. We demonstrate numerically that the IGA solution is able to reproduce results obtained before with mixed approaches. Complex phenomena such as Marangoni-driven fingering instabilities triggered by perturbations are easily captured.

Homotopy Analysis Method for Three Types of Fractional Partial Differential Equations

In this paper, three types of fractional order partial differential equations, including the fractional Cauchy–Riemann equation, fractional acoustic wave equation, and two-dimensional space partial differential equation with time-fractional-order, are considered, and these models are obtained from the standard equations by replacing an integer-order derivative with a fractional-order derivative in Caputo sense. Firstly, we discuss the fractional integral and differential properties of several functions which are derived from the Mittag-Leffler function. Secondly, by using the homotopy analysis method, the exact solutions for fractional order models mentioned above with suitable initial boundary conditions are obtained. Finally, we draw the computer graphics of the exact solutions, the approximate solutions (truncation of finite terms), and absolute errors in the limited area, which show that the effectiveness of the homotopy analysis method for solving fractional order partial differential equations.

Линейные обратные задачи для вырожденного эволюционного уравнения с производной Герасимова–Капуто в секториальном случае

Исследуется однозначная разрешимость линейных обратных задач с независящим от времени неизвестным коэффициентом для эволюционного уравнения в банаховом пространстве с вырожденным оператором при дробной производной Герасимова-Капуто. Предполагается, что пара операторов в уравнении (при дробной производной и при искомой функции) порождает разрешающее семейство операторов соответствующего вырожденного линейного однородного уравнения дробного порядка. Исходная задача редуцирована к системе двух задач: задача для алгебраического уравнения на подпространстве вырождения исходного уравнения и задача для уравнения, разрешенного относительно дробной производной, на дополнении к подпространству вырождения. Продемонстрировано два подхода. Первый из них предполагает исследование обратной задачи для разрешенного относительно производной уравнения и прямой задачи для алгебраического уравнения. При осуществлении второго подхода сначала исследуется обратная задача для уравнения на подпространстве вырождения, после чего исследуется прямая задача для второго уравнения. Абстрактные результаты использованы при исследовании начально-краевых задач для одного класса уравнений в частных производных дробного порядка по времени с неизвестным коэффициентом, зависящим от пространственных переменных. We investigate the unique solvability of linear inverse problems for the evolution equation in a Banach space with a degenerate operator at the fractional Gerasimov-Caputo derivative and with a time-independent unknown coefficient. It is assumed that a pair of operators in the equation (at the unknown function and at its fractional derivative) generates a family of resolving operators of the corresponding degenerate linear homogeneous equation of the fractional order. The original problem is reduced to a system of two problems: the problem for an algebraic equation on the degeneration subspace of the original equation and the problem for the equation solved with respect to the fractional derivative, on the complement to the degeneration subspace. Two approaches are demonstrated. The first involves the study of the inverse problem for the equation solved with respect to the derivative and the direct problem for the algebraic equation. In the second approach, the inverse problem for the equation on the degeneration subspace is investigated firstly, then the direct problem for the second equation is researched. Abstract results are used to study initial-boundary value problems for a class of time-fractional order partial differential equations with an unknown coefficient depending on the spatial variables.

Behaviour under Moving Distributed Masses of Simply Supported Orthotropic Rectangular Plate Resting on a Constant Elastic Bi-Parametric Foundation

This work investigates the behavior under Moving distributed masses of orthotropic rectangular plates resting on bi-parametric elastic foundation. The governing equation is a fourth order partial differential equation with variable and singular co-efficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al[1]. This is then simplified using modified asymptotic method of Struble. The closed form solution is analyzed, resonance conditions are obtained and the results are presented in plotted curves for both cases of moving distributed mass and moving distributed force.

A new family of orthogonal polynomials in three variables

In this paper we introduce a six-parameter generalization of the four-parameter three-variable polynomials on the simplex and we investigate the properties of these polynomials. Sparse recurrence relations are derived by using ladder relations for shifted univariate Jacobi polynomials and bivariate polynomials on the triangle. Via these sparse recurrence relations, second order partial differential equations are presented. Some connection relations are obtained between these polynomials. Also, new results for the four-parameter three-variable polynomials on the simplex are given. Finally, some generating functions are derived.

Computational Analysis for Unsteady and Steady Magnetohydrodynamic Radiating Nano Fluid Flow past a Slippery Stretching Sheet Immersed in a Permeable Medium

This work gives an account of unsteady and steady radiating magnetohydrodynamics (MHD) nanofluid flow over a slippery stretching sheet embedded in a porous medium. A widely used similarity variables of such flows reduces a governing partial differential equations (PDEs) into a new set of partial differential equations (PDEs) in which a dependent variable is a function of two independent variables rather than three. At the same time, third order PDEs are converted into a second order PDEs by defining a new variable wherever it is deemed necessary. For the time integration, we perform first order explicit Euler method and spatial derivatives are approximated by the finite difference formulas in the context of node-centered finite volume method. It is revealed that for the unsteady flow the temperature of the nanofluid is higher near the surface without thermophoresis parameter Nt and reduced significantly when Nt is present. We also show that the concentration boundary layer thickness decreases with an increase of Darcy number Da.

Non-local initial problem for second order time-fractional and space-singular equation

In this work, we consider an initial problem for second order partial differential equations with Caputo fractional derivatives in the time-variable and Bessel operator in the space-variable. For non-local boundary conditions, we present a solution of this problem in an explicit form representing it by the Fourier-Bessel series. The obtained solution is written in terms of multinomial Mittag-Leffler functions and first kind Bessel functions.

Development of an Equivalent Circuit for Batteries Based on a Distributed Impedance Network

This paper presents an original equivalent battery model based on a distributed RC circuit. The model virtually includes an infinite number of time constants to reflect the battery behavior, especially during the dynamic stress test. The model has much fewer parameters than the traditional battery model and so reduces the parameterization effort significantly. The self-discharge is inherently implemented in the model as well. The second order partial differential equation of the model is derived and both explicit and numerical solution approaches are introduced to incorporate the model in any simulation environment or analytical response analysis. Incorporating the effect of the state of charge on the parameters of this model is more easily accomplished by changing the trends of the parameters as opposed to each of them individually. The model matches to various battery type and it is validated on a 30 Ah Lithium-Ion battery from 1 C to 8 C discharge pulse current. Furthermore, the results are compared to the classic battery model.

A Theoretical Method to Compare Relative Bulk Thermal Sensitivities of Several Common High Explosives

AbstractSimultaneous numerical solutions of the second and first order partial differential equations describing the first order decomposition of seven explosives are presented. The solutions were used to compare the critical rates of heat production per unit mass of each material. The lower such a rate is the more sensitive is the explosive. The relative sensitivities of Tetryl, PETN, RDX, TNT, HMX and TATB are as might have been predicted. However, PBX9404 shows apparently enhanced thermal sensitivity compared with these six explosives. This provides a warning that some modern low vulnerability composite explosives based partly on propellant ingredients might constitute enhanced thermal threats.

Solution of Biharmonic Equation in Complicated Geometries With Physics Informed Extreme Learning Machine

Abstract Recently, physics informed neural networks (PINNs) have produced excellent results in solving a series of linear and nonlinear partial differential equations (PDEs) without using any prior data. However, due to slow training speed, PINNs are not directly competitive with existing numerical methods. To overcome this issue, the authors developed Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINN, and tested it on a range of linear PDEs of first and second order. In this paper, we evaluate the effectiveness of PIELM on higher-order PDEs with practical engineering applications. Specifically, we demonstrate the efficacy of PIELM to the biharmonic equation. Biharmonic equations have numerous applications in solid and fluid mechanics, but they are hard to solve due to the presence of fourth-order derivative terms, especially in complicated geometries. Our numerical experiments show that PIELM is much faster than the original PINN on both regular and irregular domains. On irregular domains, it also offers an excellent alternative to traditional methods due to its meshless nature.

A fictitious points one–step MPS–MFS technique

The method of fundamental solutions (MFS) is a simple and efficient numerical technique for solving certain homogenous partial differential equations (PDEs) which can be extended to solving inhomogeneous equations through the method of particular solutions (MPS). In this paper, radial basis functions (RBFs) are considered as the basis functions for the construction of a particular solution of the inhomogeneous equation. A hybrid method coupling these two methods using both fundamental solutions and RBFs as basis functions has been effective for solving a large class of PDEs. In this paper, we propose an improved fictitious points method in which the centres of the RBFs are distributed inside and outside the physical domain of the problem and which considerably improves the performance of the MPS–MFS. We also describe various techniques to deal with the several parameters present in the proposed method, such as the location of the fictitious points, the source location in the MFS, and the estimation of a good value of the RBF shape parameter. Five numerical examples in 2D/3D and for second/fourth–order PDEs are presented and the performance of the proposed method is compared with that of the traditional MPS–MFS.

FORCED RESPONSE VIBRATION OF SIMPLY SUPPORTED BEAMS WITH AN ELASTIC PASTERNAK FOUNDATION UNDER A DISTRIBUTED MOVING LOAD

In this study, the response of two homogeneous parallel beams with two-parameter Pasternak elastic foundation subjected to a constant uniform partially distributed moving force is considered. On the basis of Euler-Bernoulli beam theory, the fourth order partial differential equations of motion describing the behavior of the beams when subjected to a moving force were formulated. In order to solve the resulting initial-boundary value problem, finite Fourier sine integral technique and differential transform scheme were employed to obtain the analytical solution. The dynamic responses of the two beams obtained was investigated under moving force conditions using MATLAB. The effects of speed of the moving force, layer parameters such as stiffness (K_0) and shear modulus (G_0 ) have been conducted for the moving force. Various values of speed of the moving load, stiffness parameters and shear modulus were considered. The results obtained indicates that response amplitudes of both the upper and lower beams increases with increase in the speed of the moving load. Increasing the stiffness parameter is observed to cause a decrease in the response amplitudes of the beams. The response amplitudes decreases with increase in the shear modulus of the linear elastic layer.

Solvability of Impulsive Nonlinear Partial Differential Equations with Nonlocal Conditions

The aim of this paper is to investigate the theoretical approach for solvability of impulsive abstract Cauchy problem for impulsive nonlinear fractional order partial differential equations with nonlocal conditions, where the nonlinear extensible beam equation is a particular application case of this problem.

The LMAPS for solving fourth-order PDEs with polynomial basis functions

Due to certain difficulties in solving fourth-order partial differential equations (PDEs) using localized methods, the given differential equation is normally split into two decoupled second order PDEs. Such an approach is only feasible for Dirichlet and Laplace boundary conditions. In this paper the localized method of particular solutions is applied to fourth-order PDEs directly using polynomial basis functions. The effectiveness of the proposed algorithms is demonstrated by considering four numerical examples.

Application of New Iterative Method to Time Fractional Whitham–Broer–Kaup Equations

This article presents the Fractional Laplace transform with the help of new iterative method for finding the approximate solutions of coupled system of fractional order partial differential equations. The time fractional Whitham-Broer-Kaup system is taken as test example The fractional derivatives are described in the Caputo sense. Numerical results obtained by the proposed method are compared with that of Adomian Decomposition Method (ADM), Variational Iteration Method (VIM) and Optimal Homotopy Asymptotic Method (OHAM).Numerical results show that the proposed method is reliable and efficient for solution of fractional order coupled system partial differential equations. The accuracy of the method increases by taking higher order approximations.

Uniqueness of the solution of nonlinear singular first order partial differential equations

This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t = F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. Under a very weak assumption we show the uniqueness of the solution of this equation. The results are applied to the problem of analytic continuation of local holomorphic solutions of equations of this type.

The spreading of an axisymmetric ion beam by space charge and diffusion

An analysis is presented of the evolution of a narrow ion beam, produced for instance by an electrospray source, in ambient pressure air. To minimize beam broadening, an intense axial electric field E is externally imposed. Still, radial beam broadening due to the ionic diffusivity D and space charge must be accounted for. Ignoring axial contributions of diffusion and space charge, a simple second order partial differential equation is obtained for the space charge induced field. In the planar problem, this field is governed by Burgers's equation, which is exactly solvable via the Cole-Hopf transformation. The axisymmetric problem is not similarly tractable by any familiar transformation, but may be reduced to an ordinary differential equation in terms of a similarity variable, yielding the complete beam structure under all regimes of interest. The beam broadening effect of space charge is governed by the ratio between the beam current I and the group 2πε0DE, where ε0 is the permittivity of the gas.

Efficient Design of Neural Networks for Solving Third Order Partial Differential Equations

The aim of the research is design efficient neural networks to solve important type of partial differential equations represent many important models. The efficiency of design based on chooses a suitable activation function, suitable training algorithms with study how to increase the speed of this algorithm. Then we used suggested design to solve third order PDEs.

Modeling the Time Delay Problem of Galactic Cosmic Ray Flux in Solar Cycles 21 and 23

We present a 2-dimensional time-dependent simulation based on Parker’s transport equation (PTE) describing the propagation of energetic astroparticles – cosmic rays (CR) in the heliosphere. PTE is a second order partial differential equation containing four major processes: diffusion, convection, drift, and adiabatic cooling responsible for modulation of the CR flux in the heliosphere. We implement in the numerical simulation, a few physical parameters as the tilt angle, delta , of the heliospheric current sheet (HCS), module, B, of the interplanetary magnetic field (IMF), and variations of drift effect of the CR particles with solar activity (SA). Our approach is the implementation of two independent parameters (proxies), gamma and nu . The parameters gamma and nu are calculated from various independent sources, gamma , from neutron monitors (NM) daily data, and nu , using the IMF hourly data. The solutions of PTE obtained from our numerical model are compared with the variations of the CR flux recorded by NMs. We prove the existence of a varying delay time (DT) between the changes of galactic cosmic ray (GCR) intensity and the parameters characterizing SA. Based on our investigation, we obtained different DTs in Solar Cycles 21 and 23. We conclude that the calculated DTs, after comparison with the observed DTs, are useful parameters for the study of GCR transport in the heliosphere.