Higher-order Differential Equations Research Articles

Using Laplace Residual Power Series Method in Solving Coupled Fractional Neutron Diffusion Equations with Delayed Neutrons System

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.

Numerical solution of neutral delay differential equations using orthogonal neural network

In this paper, an efficient orthogonal neural network (ONN) approach is introduced to solve the higher-order neutral delay differential equations (NDDEs) with variable coefficients and multiple delays. The method is implemented by replacing the hidden layer of the feed-forward neural network with the orthogonal polynomial-based functional expansion block, and the corresponding weights of the network are obtained using an extreme learning machine(ELM) approach. Starting with simple delay differential equations (DDEs), an interest has been shown in solving NDDEs and system of NDDEs. Interest is given to consistency and convergence analysis, and it is seen that the method can produce a uniform closed-form solution with an error of order 2^{-n}, where n is the number of neurons. The developed neural network method is validated over various types of example problems(DDEs, NDDEs, and system of NDDEs) with four different types of special orthogonal polynomials.

Higher-Order Delay Differential Equation with Distributed Deviating Arguments: Improving Monotonic Properties of Kneser Solutions

This study aims to investigate the oscillatory behavior of the solutions of an even-order delay differential equation with distributed deviating arguments. We first study the monotonic properties of positive decreasing solutions or the so-called Kneser solutions. Then, by iterative deduction, we improve these properties, which enables us to apply them more than once. Finally, depending on the symmetry between the positive and negative solutions of the studied equation and by combining the new condition for the exclusion of Kneser solutions with some well-known results in the literature, we establish a new standard for the oscillation of the investigated equation.

Improved Properties of Positive Solutions of Higher Order Differential Equations and Their Applications in Oscillation Theory

In this article, we present new criteria for testing the oscillation of solutions of higher-order neutral delay differential equation. By deriving new monotonic properties of a class of the positive solutions of the studied equation, we establish better criteria for oscillation. Furthermore, we improve these properties by giving them an iterative character, allowing us to apply the criteria more than once. The results obtained in this paper are characterized by the fact that they do not require the existence of unknown functions and do not need the commutation condition to composition of the delay functions, which are necessary conditions for the previous related results.

Multipoint boundary value problems for higher-order Hadamard fractional neutral differential equations and inclusions

In this paper, we consider fractional neutral differential equations with multipoint boundary value conditions involving Hadamard derivatives and integrals. We obtain the existence and uniqueness of the solution of the equation by using several fixed point theorems, and we also consider the Ulam–Hyers stability of the solution. In addition, we study the differential inclusion problem with multipoint boundary value conditions and prove the existence of the solution of the boundary value problem when the multivalued map has convex values. We also give several examples to illustrate the feasibility of the results.

Solving System of Higher-Order Linear Differential Equations On The Level of Operators

In this paper, I present a method for solving the system of higher-order linear differential equations (HLDEs) with inhomogeneous initial conditions on the level of operators. Using proposed method, we compute the matrix Green’s operator as well as the vector Green’s function of a fully-inhomogeneous initial value problems (IVPs). Examples are discussed to demonstrate the proposed method.

Channel Prediction Using Ordinary Differential Equations for MIMO Systems

Channel state information (CSI) estimation is part of the most fundamental problems in 5G wireless communication systems. In mobile scenarios, outdated CSI will have a serious negative impact on various adaptive transmission systems, resulting in system performance degradation. To obtain accurate CSI, it is crucial to predict CSI at future moments. In this paper, we propose an efficient channel prediction method in multiple-input multiple-output (MIMO) systems, which combines genetic programming (GP) with higher-order differential equation (HODE) modeling for prediction, named GPODE. In the first place, the variation of one-dimensional data is depicted by using higher-order differential, and the higher-order differential data is modeled by GP to obtain an explicit model. Then, a definite order condition is given for the modeling of HODE, and an effective prediction interval is given. In order to accommodate to the rapidly changing channel, the proposed method is improved by taking the rough prediction results of Autoregression (AR) model as a priori, i.e., Im-GPODE channel prediction method. Given the effective interval, an online framework is proposed for the prediction. To verify the validity of the proposed methods, We use the data generated by the Cluster Delay Line (CDL) channel model for validation. The results show that the proposed methods has higher accuracy than other traditional prediction methods.

Computation of Cattaneo-Christov heat and mass flux model in Williamson nanofluid flow with bioconvection and thermal radiation through a vertical slender cylinder

Nanofluids are in high demand in the medical, electrical, and engineering fields. Nanofluid flow is used in a variety of applications including industrial cooling and heating frameworks, hyperthermia, electronic device batteries, and related pharmaceutical administration systems. Because of their improved thermal properties, nanofluids are commonly used as coolants in heat exchangers also including heating systems, electronic cooling systems, and radiators. Many scientists have studied heat transfer via a cylinder. The current study uses a vertical thin cylinder to scrutinize the effects of activation energy, motile microorganisms, bioconvection, and WU's slip on the Williamson nanofluid. Brownian motion, thermal radiation, Cattaneo-Christov heat, mass flow, and thermophoresis are all considered. Using appropriate similarity variables, the structures of PDEs are translated into the structure of ordinary differential equations. Using the well-known shooting strategy (Bvp4c) in the mathematical program MATLAB, higher-order nonlinear differential equations (DEs) are reduced to first-order. The effects of important flow parameters are shown graphically and numerically. It is noted that the momentum distribution field is upsurged for mixed convection parameters and decreased for the fluid parameter. The thermal profile is declining for the Prandtl number while increasing for the thermal radiation parameter and heat source-sink parameter. It is also observed that the concentration distribution field decreased for the Lewis number and concentration relaxation parameter while boosted up for larger magnitude of activation energy and thermophoresis parameter. It is analyzed that the microorganisms profile is upsurged for bioconvection Rayleigh number while declined for Peclet number. A comparison with the reported results from available literature is also done, and an excellent agreement is found. The discovery has far-reaching ramifications in the fields of medicine, modern aircraft technology, and power generation. In this study, we combine metallic nanoparticles with the base fluid water. Metallic nanoparticles offer a wide range of uses in everyday life, including catalysts, biosensors, medicine, and medication delivery, soil pollutant removal, electronics, coatings, packaging, semiconductors, bioengineering, paints, telecommunications, cosmetics, automobiles, and water treatment.

Oscillation of higher-order canonical delay differential equations: comparison theorems

In this work, we study the oscillatory properties of a higher-order neutral delay differential equation. By using the principle of comparison with equations of the first order we establish a set of new oscillation criteria for this equation under the canonical condition. Furthermore, the new criteria extend and complement some previous results in the literature. To that end, we compare these criteria by applying them to special cases of the equations under consideration in order to determine which one is the most efficient and least restrictive.

The Exact WKB analysis for asymmetric scalar preheating

Using the exact WKB analysis of the higher-order differential equations, we analyze the asymmetry in dynamical particle production of a complex scalar field. The solution requires the Stokes phenomena of the fourth-order differential equation. We found that the interference of different types of the Stokes phenomena causes matter-antimatter asymmetry. We also showed a specific example where asymmetry is forbidden in the exact calculation, but a false asymmetry appears in the perturbative expansion.

Micro-polar nanofluid in the presence of thermophoresis, hall currents, and Brownian motion in a rotating system

This study examined a rotating system by a micro-polar nanofluid between two parallel plates in the presence of magnetic and electric fields. The flow study has been performed in a steady-state. The governing equations of the present problem are transformed into nonlinear and coupled equations with appropriate similarity variables. The impacts of the Nusselt number, skin friction, and Sherwood number on temperature, velocity, and concentration distribution have been discussed. This research has mainly investigated the effects of the rotation, Brownian motion, thermophoresis analysis, and Hall current of micro-polar nanofluid. Results demonstrate for weak concentration [Formula: see text] and strong concentration [Formula: see text], that Nusselt number (Nu) increased with higher value of [Formula: see text], [Formula: see text] and decreased when [Formula: see text], Sc and [Formula: see text] increased. Also, by increasing the [Formula: see text] and Sc numbers, the temperature profile is decreased and increased by increasing [Formula: see text], [Formula: see text], and [Formula: see text]. In addition, at a higher value of the [Formula: see text], the velocity profile in the [Formula: see text]-direction increased because of increasing the fluid motion and by fast ionization, increasing the [Formula: see text] parameter raised the velocity profile in the [Formula: see text]-direction. There was a decrease in the velocity profile in the [Formula: see text]-direction and the micro-rotation velocity profile. Our results show that the method used is very efficient and practical for solving this category of coupled equations, and that the solution of higher-order nonlinear differential equations in engineering is very consistent. Also, by comparing the obtained results with the previous results, the obtained values differ by about 6%.

The Onsager–Machlup theory of fluctuations and time-dependent generalized normal distribution

Abstract Generalization of the variational formulation of the Onsager–Machlup thermodynamic theory of fluctuation is considered. Within the framework of variational theory, we introduce the time-dependent generalized normal distribution and Hamilton–Jacobi equation. The family of higher-order partial differential equations, which generalize classical Fokker–Planck equation, is considered. It is shown that proposed theory can be used for describing anomalous diffusion.

Linear higher-order fractional differential and integral equations

We study the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L1(a,b) and C[a,b] when n≥ 2. We establish the equivalence in C[a,b] of the IVP of the nth order FDE subject to the initial condition u(i)(a)=ui for i in {0,1,...,n-1} when n≥2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n≥2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n≥2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory.

Connection of Balancing numbers with solution of a system of two higher-order difference equations

Connection of Balancing numbers with solution of a system of two higher-order difference equations

Some generalized numerical methods for solving higher-order of fractional partial differential equations with application

This article recommends a computer method for solving some classes of higher-order fractional partial differential equations (FrPDEs). The updated techniques were made by combining the method of lines (MOL) and the numerical RK-type method. For solving certain types of FrPDEs that aren’t in the order, you might expect. Some problems with second-, third-, and fourth-order FrPDEs are solved with the methods made, and the numerical solutions are compared with the analytical answers. Also, a new type of application in classical physics called a wave FrPDE has been made for mechanical waves like water, earthquake, sound, and electromagnetic waves. Many examples are given to show how flexible and useful the method is. The numerical results are seen as the same as the exact solutions of the implementations to show how accurate and useful the suggested methods are. Also, the numerical implementations compare the suggested methods with exact solutions to show that they work and are accurate. The suggested methods used RKM analytical solutions to come up with numerical solutions to the test problems. Some examples are given to show how flexible and useful the method is. Also, the numerical implementations are compared with exact results to show that the proposed methods work and are accurate. R.K. techniques were used to find numerical answers to the suggested test problems.

NEW OSCILLATION CRITERIA FOR A CLASS OF HIGHER-ORDER NEUTRAL FUNCTIONAL DYNAMIC EQUATIONS ON TIME SCALES

In this paper, we study the higher-order neutral functional dynamic equations of the form \begin{equation*} L_ny(t)+q(t)f(|y(\theta(t))|^{\beta}sgn(y(\theta(t))))=0,\ \ t\in[t_0,\infty)_\mathbb{T}, \end{equation*} on an arbitrary time scale $\mathbb{T}$ with $\sup\mathbb{T}=\infty$, where \begin{equation*} L_1y(t)=[y(t)+r(t)y(\tau(t))]^{\Delta},\ L_{i+1}y(t)=[p_i(t)|L_iy(t)|^{\alpha_i}sgn(L_iy(t))]^{\Delta}, \end{equation*} $\alpha_i$, $1\leq i\leq n-1$ and $\beta$ are positive constants, $p_i$, $1\leq i\leq n-1$ and $q$ are rd-continuous functions from $[t_0,\infty)_{\mathbb{T}}$ to $[0,\infty)$ and $r\in \mathrm{C}_{\mathrm{rd}}(\mathbb{T},[0,1))$. The functions $\tau,\theta\in \mathrm{C}_{\mathrm{rd}}(\mathbb{T},\mathbb{T})$ satisfy $\tau(t)\leq t$ and $\lim_{t\rightarrow\infty}\tau(t)=\lim_{t\rightarrow\infty}\theta(t)=\infty$. Criteria are established for the oscillation of solutions for both even and odd order cases. The obtained results here generalize and improve some known results for oscillation of the corresponding higher-order ordinary differential equations [?], but the proof of these counterparts are quite different from the literature. Finally, some interesting examples are given to illustrate the versatility of our main results.

The new general solution for a class of fractional-order impulsive differential equations involving the Riemann-Liouville type Hadamard fractional derivative

<abstract><p>In this paper, the new general solution for a class of higher-order impulsive fractional differential equations (IFDEs) involving the Riemann-Liouville (R-L) type Hadamard fractional derivative (FD) is presented. Specifically, the necessary and sufficient conditions of the solution are obtained by converting boundary value problems (BVPs) into integral equations and applying analytical techniques. The results in the paper provide a new method for converting BVPs or initial value problems (IVPs) for IFDEs to integral equations. Finally, some examples are devoted to explaining the application of the theorem.</p></abstract>

Analysis of a Squeezing Flow of a Casson Nanofluid between Two Parallel Disks in the Presence of a Variable Magnetic Field

The present article deals with the MHD flow of a Casson nanofluid between two disks. The lower disk was fixed as well as permeable. The upper disk was not permeable, but it could move perpendicularly up and down toward the lower disk. Titanium dioxide was selected as nanoparticles and water as a base fluid. The governing higher-order nonlinear partial differential equations were transformed into a set of nonlinear ordinary differential equations by using similarity transformation. The differential transform method (DTM) was applied to solve the nonlinear ODEs. The nature of the velocity profiles for the different values of the suction injection parameter, the squeeze number, the Casson fluid parameter, and the volume fraction parameter of the nanofluid are pictorially discussed in this paper. The coefficient of skin friction was tabulated for the novelty of the research. The comparison of the results was determined by the DTM and the numerical methods. The profile values were also compared with the literature work and found to agree. This comparative study proves the accuracy and efficiency of the method. It is concluded from this research that the flow properties behave oppositely for all parameters during suction and injection.

Numerical simulations of a mixed finite element method for damped plate vibration problems

<abstract><p>The mixed finite element method can reduce the requirement for the smoothness of the finite element space and simplify the interpolation space for finite elements, and hence is especially effective in solving high order differential equations. In this work, we establish a mixed finite element scheme for the initial boundary conditions of damped plate vibrations and prove the existence and uniqueness of the solution of the semi-discrete and backward Euler fully discrete schemes. We use linear element approximation for the introduced intermediate variables, conduct the error analysis, and obtain the optimal order error estimate. We verify the efficiency and the accuracy of the mixed finite element scheme via numerical case studies and quantify the influence of the damping coefficient on the frequency and amplitude of the vibration.</p></abstract>

Periodic solutions for a higher-order p-Laplacian neutral differential equation with multiple deviating arguments

In this article, we consider the following high-order p-Laplacian neutral differential equation with multiple deviating arguments:$$(\varphi_{p}(x(t)-cx(t-r))^{(m)}(t)))^{(m)}= f(x(t))x'(t)+g(t,x(t),x(t-\tau_{1}(t)),...,x(t-\tau_{k}(t)))+e(t).$$By appling the continuation theorem, theory of Fourier series, Bernoulli numbers theory and some analytic techniques, sufficient conditions for the existence of periodic solutions are established.