Technique For Differential Equations Research Articles

An Efficient Analytical Technique for Time-Fractional Parabolic Partial Differential Equations

In this work, we examine the time-fractional fourth-order parabolic partial differential equations with the aid of the optimal homotopy asymptotic method (OHAM). The 2nd order approximate results obtained by using the suggested scheme are compared with the exact solution. It has been noted that the results achieved via OHAM have a large convergence rate for the problems. The solutions are graphically analyzed, and the relative errors are presented in tabular form.

Riccati Technique and Asymptotic Behavior of Fourth-Order Advanced Differential Equations

In this paper, we deal with the oscillation of fourth-order nonlinear advanced differential equations of the form r t y ‴ t α ′ + p t f y ‴ t + q t g y σ t = 0 . We provide oscillation criteria for this type of equations, and examples to illustrate the criteria.

Approximate impedance for time-harmonic Maxwell's equations in a non planar domain with contrasted multi-thin layers

The aim of this paper is to give asymptotic models for the impedance of contrasted multi-thin layers for the harmonic Maxwell's equations. We start from a transmission problem which describes the scattering of electromagnetic waves by an obstacle covered with a thin coating (superposition of different thin layers of dielectric materials). We show how to model the effect of the thin coating by an impedance boundary condition on the boundary of the propagation domain. To this end, we use a technique of abstract differential equations.

Circuit Modelling by Difference Equation: Pedagogical Advantages and Perspectives

Circuit theory is a cornerstone course in electrical engineering and control majors in ordinary universities and colleges throughout the world. This course covers fundamental principles and analysis methods of basic circuits commonly employed in the forthcoming courses. In most electrical programs after the introduction of basic elements of Ohm's and Kirchhoff's current and voltage laws, the dynamic response of the circuits containing capacitors and inductors will be studied. Customarily to solve these circuits, advanced mathematical approaches such as differential equations are used. Under such circumstances, the students are faced with two challenges, solving the differential equations, and understanding the dynamic response of circuits. In order to improve students' understanding, an analysis tool with less mathematical prerequisites should be used for the solutions before embarking on the use of conventional differential equation techniques such as Laplace transform. Hence, we propose a novel approach for these circuit analyses through the application of a discretized version of differential equations which is used in discrete control systems. Although this approach has a well-established background, its exploration uses in the circuit theory course as yet has not been reported. The novelty of the proposed approach not only lies in its intuitive simplicity but also in its contribution to the understanding and visualization of students in the real-time response of linear and non-linear circuits to any desirable input without any mathematical burden. The analysis can be performed by hand or this is also helpful for those who prefer modern education aided by computers. This, in turn, may attract more students to the program. In this paper, the effectiveness of the proposed approach is demonstrated through a set of illustrative examples.

The use of the apparatus of ordinary differential equations in simulation of economic and environmental systems

The ordinary differential equations techniques applying to investigate the economical and ecological systems has been considered in presented article. The interconnected economical complexes development for the countries with the different economical potential has been simulated. The population economical activity influence on the environment pollution and the state of region’s flora has been investigated. The economical efficiency of the new technical diagnostics implementation has been studied. The methods of presented models realization has been presented and investigated, the results of tested calculations have been presented and one’s analysis has been given. The directions of future investigations have been determined.

Analytical technique for neutral delay differential equations with proportional and constant delays

Neutral delay differential equations (NDDEs) are a type of delay differential equations (DDEs) that arise in numerous areas of applied sciences and play a vital role in mathematical modelling of real-life phenomena. Some techniques have experienced difficulties in finding the approximate analytical solution which converges rapidly to the exact solution of these equations. In this paper, an analytical approach is proposed for solving linear and nonlinear NDDEs with proportional and constant delays based on the homotopy analysis method (HAM) and natural transform method where the nonlinear terms are simply calculated as a series of, He’s polynomial. The proposed method produces solutions in the form of a rapidly convergent series which leads to the exact solution from only a few numbers of iterations. Some illustrative examples are solved, and the convergence analysis of the proposed techniques was also provided. The obtained results reveal that the approach is very effective and efficient in handling both linear and nonlinear NDDEs with proportional and constant delays and can also be used in various types of linear and nonlinear problems.

A new class of A-stable numerical techniques for ordinary differential equations: Application to boundary-layer flow

The present attempt is made to propose a new class of numerical techniques for finding numerical solutions of ODE. The proposed numerical techniques are based on interpolation of a polynomial. Currently constructed numerical techniques use the additional information(s) of derivative(s) on particular grid point(s). The advantage of the presently proposed numerical techniques is that these techniques are implemented in one step and can provide highly accurate solution and can be constructed on fewer amounts of grid points but has the disadvantage of finding derivative(s). It is to be noted that the high order techniques can be constructed using just two grid points. Presently proposed fourth order technique is A-stable but not L-stable. The order and maximum absolute error are found for a fourth order technique. The fourth order technique is employed to solve the Darcy-Forchheimer fluid-flow problem which is transformed further to a third-order non-linear boundary value problem on the semi-infinite domain.

Numerical solutions of hybrid fuzzy differential equations in a Hilbert space

The main goal of this work is to study a numerical method for certain hybrid fuzzy differential equations with an application of a reproducing kernel Hilbert space technique for fuzzy differential equations. Meanwhile, we construct a system of orthogonal functions of the space W22[a,b]⊕W22[a,b] depending on a Gram-Schmidt orthogonalization process to get approximate-analytical solutions of a hybrid fuzzy differential equation. A proof of convergence of this method is also discussed in detail. The exact as well as the approximate solutions are displayed by a series in terms of their α-cut representation form in the Hilbert space W22[a,b]⊕W22[a,b]. To demonstrate behavior, efficiency, and appropriateness of the present technique, two different numerical experiments are solved numerically in this paper.

Event-Triggered Average Consensus of Multiagent Systems with Switching Topologies

Event-triggered average consensus of multiagent systems with switching topologies is studied in this paper. A distributed protocol based on event-triggered time sequences and switching time sequences is designed. Based on the inequality technique and stability theory of differential equations, a sufficient condition for achieving average consensus is obtained under the assumption that switching signal is ergodic and the total period over which connected topologies is sufficiently large. A numerical simulation is presented to show the effectiveness of the theoretical results.

Approximating Solutions of Non Linear First Order Abstract Measure Differential Equations by Using Dhage Iteration Method

In this paper we have proved the approximating solutions of the nonlinear first order abstract measure differential equation by using Dhage’s iteration method. The main result is based on the iteration method included in the hybrid fixed point theorem in a partially ordered normed linear space. Also we have solved an example for the applicability of given results in the paper. Sharma [2] initiated the study of nonlinear abstract differential equations and some basic results concerning the existence of solutions for such equations. Later, such equations were studied by various authors for different aspects of the solutions under continuous and discontinuous nonlinearities. The study of fixed point theorem for contraction mappings in partial ordered metric space is initiated by different authors. The study of hybrid fixed point theorem in partially ordered metric space is initiated by Dhage with applications to nonlinear differential and integral equations. The iteration method is also embodied in hybrid fixed point theorem in partially ordered spaces by Dhage [12]. The Dhage iteration method is a powerful tool for proving the existence and approximating results for nonlinear measure differential equations. The approximation of the solutions are obtained under weaker mixed partial continuity and partial Lipschitz conditions. In this paper we adopted this iteration method technique for abstract measure differential equations.

The cost of controlling strongly degenerate parabolic equations

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).

Numerical study of an influenza epidemic dynamical model with diffusion

In this paper, a deterministic model is formulated in the aim of performing a thorough investigation of the transmission dynamics of influenza. The main advantage of our model compared to existing models is that it takes into account the effects of hospitalization as well as the diffusion. The proposed model consisting of a dynamical system of partial differential equations with diffusion terms is numerically solved using fast and accurate numerical techniques for partial differential equations. Furthermore, the basic reproduction number that guarantees the local stability of disease-free steady state without diffusion term is calculated. Various numerical simulation for different values of the model input parameters are finally presented in order to show the effect of the effective contact rate on the steady state of the different population compartments.

Monotone Iterative Technique for Nonlocal Impulsive Finite Delay Differential Equations of Fractional Order

The paper is concerned with the extension of a monotone iterative technique to impulsive finite delay differential equations of fractional order with a nonlocal initial condition in an ordered Banach space. We study the existence of extremal mild solutions with or without assuming the compactness of a semigroup and also prove the uniqueness of the mild solution of the system. The results are obtained with the help of fractional calculus, a measure of non-compactness, the semigroup theory and monotone iterative technique. Finally, an example is provided to show the application of our main.

Combining standard with optimised split‐step finite‐difference time‐domain methods for the study of graphene configurations

In this study, the authors propose a hybrid computational model for the reliable simulation of electromagnetic-wave phenomena emerging in graphene structures, which incorporates two variations of an unconditionally-stable finite-difference time-domain algorithm. The new approach features: (a) a dispersive model that relies on the auxiliary differential equation technique and takes into consideration the graphene's surface conductivity, and (b) a modified discretisation algorithm implementing error-optimised spatial approximations, which is appropriate for graphene-free mesh nodes. The resulting update equations are devoid of time-step restrictions, thus formulating a suitable and useful computational framework for the investigation of contemporary graphene-based problems, which commonly call for densely-sampled grids and/or prolonged simulations. After theoretically assessing the fundamental features of the proposed algorithm, the propagation properties of surface waves on graphene are extracted numerically and compared with analytical estimations, so that computational validation is provided. In addition, a realistic surface-wave coupler configuration is simulated and its main characteristics are extracted accurately, exemplifying the method's robustness on more complex setups.

Unified killing mechanism in a single server queue with renewal input

Queueing systems experienced in real-life situations are very often influenced by negative arrivals which are independent of service process and cause the elimination of jobs from the system. Such a scenario occurs in computer network and telecommunication systems where an attack by a malicious virus results in the removal of some or all data files from the system. Along this direction many authors have proposed various killing processes in the past. This paper unifies different killing mechanisms into the classical single server queue having infinite capacity, where arrival occurs as renewal process with exponential service time distribution. The system is assumed to be affected by negative customers as well as disasters. The model is investigated in steady-state in a very simple and elegant way by means of supplementary variable and difference equation technique. The distribution of system-content for the positive customers is derived in an explicit form at pre-arrival and random epochs. The influence of different parameters on the system performance are also examined.

Analytical solution of non-isothermal two-dimensional general rate model of liquid chromatography

A non-isothermal two-dimensional general rate model is formulated and analytically solved to analyze the effects of temperature changes inside liquid chromatographic columns of cylindrical geometry. The model equations form a system of convection-diffusion partial differential equations. The finite-Hankel transformation, the Laplace transformation, the eigen-decomposition technique and a conventional solution technique of ordinary differential equations are used to solve the equations of the model. The coupling between concentration and temperature fronts is demonstrated and important parameters that affect the performance of the column are evaluated. To find the ranges of validity of our analytical results, a semi-discrete high resolution finite volume method is applied to solve the same system of equations for both linear and nonlinear isotherms. The results of this contribution can be helpful to optimize non-isothermal liquid chromatographic processes in which both radial and axial gradients occur.

Mean almost periodicity and moment exponential stability of semi-discrete random cellular neural networks with fuzzy operations.

By using the semi-discretization technique of differential equations, the discrete analogue of a kind of cellular neural networks with stochastic perturbations and fuzzy operations is formulated, which gives a more accurate characterization for continuous-time models than that by Euler scheme. Firstly, the existence of at least one p-th mean almost periodic sequence solution of the semi-discrete stochastic models with almost periodic coefficients is investigated by using Minkowski inequality, Hölder inequality and Krasnoselskii’s fixed point theorem. Secondly, the p-th moment global exponential stability of the semi-discrete stochastic models is also studied by using some analytical skills and the proof of contradiction. Finally, a problem of stochastic stabilization for discrete cellular neural networks is studied.

A unified 3‐D ADI‐FDTD algorithm with one‐step leapfrog approach for modeling frequency‐dependent dispersive media

AbstractA unified 3‐D alternating‐direction‐implicit finite‐difference time‐domain algorithm with one‐step leapfrog approach is provided for analyzing the frequency‐dependent dispersive media. The dielectric parameter in the frequency domain is described by the modified Lorentz model. Simultaneously, an order reduction scheme is utilized to synchronize the auxiliary differential equations (ADE) technique with the main algorithm. The numerical cases are performed, and the results indicate that the proposed method possesses higher efficiency under allowable accuracy when compared with the analytical method and other time domain numerical schemes.

An Approximation Technique for Fractional Order Delay Differential Equations

In this research article, an Iterative Decomposition Method is applied to approximate linear and non-linear fractional delay differential equation. The method was used to express the solution of a Fractional delay differential equation in the form of a convergent series of infinite terms which can be effortlessly computable.The method requires neither discretization nor linearization. Solutions obtained for some test problems using the proposed method were compared with those obtained from some methods and the exact solutions. The outcomes showed the proposed approach is more efficient and correct.

A Numerical Method for Caputo Differential Equations and Application of High-Speed Algorithm

In this paper, a numerical algorithm to solve Caputo differential equations is proposed. The proposed algorithm utilizes the R2 algorithm for fractional integration based on the fact that the Caputo derivative of a function f(t) is defined as the Riemann–Liouville integral of the derivative f(ν)(t). The discretized equations are integer order differential equations, in which the contribution of f(ν)(t) from the past behaves as a time-dependent inhomogeneous term. Therefore, numerical techniques for integer order differential equations can be used to solve these equations. The accuracy of this algorithm is examined by solving linear and nonlinear Caputo differential equations. When large time-steps are necessary to solve fractional differential equations, the high-speed algorithm (HSA) proposed by the present authors (Fukunaga, M., and Shimizu, N., 2013, “A High Speed Algorithm for Computation of Fractional Differentiation and Integration,” Philos. Trans. R. Soc., A, 371(1990), p. 20120152) is employed to reduce the computing time. The introduction of this algorithm does not degrade the accuracy of numerical solutions, if the parameters of HSA are appropriately chosen. Furthermore, it reduces the truncation errors in calculating fractional derivatives by the conventional trapezoidal rule. Thus, the proposed algorithm for Caputo differential equations together with the HSA enables fractional differential equations to be solved with high accuracy and high speed.