Higher-order Equations Research Articles

An Investigation of the Extent of Implementation of the Financial Management Practices of Agri-SMEs in developing countries: Evidence from Tanzania

The success of agricultural Small and Medium Enterprises (agri-SMEs), among other factors, heavily relies on their ability to manage finances effectively. This study examined the extent to which agricultural SMEs in Tanzania practice financial management activities. The study employed a cross-sectional survey design to collect data from 427 agri-SMEs in three regions of Tanzania. Descriptive statistics and higher-order reflective-formative Structural Equation Model (SEM) were employed for data analysis. The findings indicated moderate adoption of financial management practices. Specifically, agri-SMEs were found to frequently engage in working capital management practices, such as maintaining sufficient cash flows and records for payables and receivables. However, financial reporting is less frequently practiced, with sporadic financial statements preparation and financial analysis. The findings indicate further that other financial management practices such as financial accounting, capital budgeting and financing management were moderately and infrequently implemented, with external financing sources being utilized less frequently compared to internal financing sources. The path analysis indicates that all lower-order constructs significantly contributed to the higher-order financial management practice construct with capital budgeting demonstrating the strongest impact. The findings highlight the need for improving financial reporting, analysis, and external financing usage in Tanzanian agri-SMEs. These findings offer valuable insights agri-SMEs in developing countries to appropriately strategize on implementation of financial management practices that will lead to improved access to finance and high performance.

Dynamics and Solutions of Higher-Order Difference Equations

The invariance method, known as Lie analysis, consists of finding a group of transformations that leave a difference equation invariant. This powerful tool permits one to lower the order, linearize and more importantly, obtain analytical solutions of difference and differential equations. In this study, we obtain the solutions and periodic solutions for some family of difference equations. We achieve this by performing an invariance analysis of this family. Eventually, symmetries are derived and used to construct canonical coordinates required for the derivation of the solutions. Moreover, periodic aspects of these solutions and the stability character of the equilibrium points are investigated.

On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations

The principle goal of this paper is to look at some of the qualitative behavior of the critical point of the rational difference equation
 
 Ψ_{n+1}=αΨ_{n-2}+((βΨ_{n-2}Ψ_{n-3})/(γΨ_{n-3}+δΨ_{n-6})), n=0,1,2,...,
 
 where α,β,γ and δ are arbitrary positive real numbers. We also used the proposed equation to get the general solution for particular cases and provided numerical examples to demonstrate our results.

The Solution of a System of Higher-Order Difference Equations in Terms of Balancing Numbers

In this paper, we are interested in the closed-form solution of the following system of nonlinear difference equations of higher order,un+1 = 1/34-vn-m , vn+1 = 1/34-un-m, n, m ∈ N0,and the initial values u-j and v-j , j∈{0, 1, ..., m} are real numbers do not equal 34. We show that the solutions of this system are associated with Balancing numbers. As consequence, these solutions are also associated with Pell numbers, Pell-Lucas numbers, and Lucas-Balancing numbers. It is shown that the global stability of positive solutions of this system holds. Our results are illustrated via numerical examples.

Numerical Analysis of Velocity and Thermal Wall Slip Effects on the Boundary Layer Flow Over an Exponentially Stretching Bullet-Shaped Object in Presence of Suction and Injection

A steady two-dimensional axisymmetric incompressible flow over an exponentially stretching bullet-shaped surface has been considered. The present work is mainly focused on fluid flow by the effect of multiple slips. The governing partial differential equations and auxiliary boundary conditions have been converted into higher-order equations by using assisting similarity transformations. These higher-order ODEs are then transformed into a 1st-order system of LDEs by the method of spectral quasi-linearization (SQLM). The validity, accuracy, and convergence of the solution have been performed by using SQLM. The fluid velocity, temperature, skin friction coefficient, and Nusselt number have been depicted graphically for the mentioned parameters as also the numerical values of velocity gradient, and Nusselt number in a table. The numerical investigation shows that the velocity gradient enhances due to the parameter of magnetic, thermal slip, and Prandtl number whereas the remaining parameters have a reverse effect on it. The heat transfer rate reduces for the parameters of magnetic, multiple slip, injection, and viscous dissipation but suction and heat generation have a reverse effect on it. The results of in this work have been justified due to the validity and accuracy of the present problem. Due to the endless application of Newtonian fluids in engineering and industry, no attempt has been taken to inspect the MHD flow with a dual slip effect along with exponential stretching bullet-shaped surface. Also, the current work is of immediate interest to those systems that are highly influenced by the heat transfer process and desired product quality.

Deep learning data-driven multi-soliton dynamics and parameters discovery for the fifth-order Kaup–Kuperschmidt equation

In this paper, we extend the multi-layer physics-informed neural networks (PINNs) to successfully learn the data-driven multi-soliton solutions and discover the coefficients of fifth-order Kaup–Kuperschmidt (KK) equation with the aid of multi-soliton data. In the forward problems, the corresponding accuracies for one-, two- and three-soliton solutions are O(10−6), O(10−4) and O(10−3), respectively. Moreover, some basic elements affecting the performance of PINNs are analyzed in detail, such as activation functions, collocation points, optimizers, neural network structures, and so on. For the inverse problems, the coefficients of the equation are discovered by the data of one-, two- and three-soliton solutions, respectively. Meanwhile, the robustness of the PINNs algorithm is explored under different noises. The accuracy of the identified coefficients can reach O(10−3) when 1% initial noise is added to the training data. And the prediction accuracy can still reach O(10−2) even if 3% initial noise is added. These numerical experiments not only show the effectiveness of PINNs to solve and discover the KK equation with some known information, but also embody the applicability to reveal the multi-soliton dynamic behaviors in higher-order nonlinear wave equations.

An application of artificial neural networks for solving fractional higher-order linear integro-differential equations

This ongoing work is vehemently dedicated to the investigation of a class of ordinary linear Volterra type integro-differential equations with fractional order in numerical mode. By replacing the unknown function by an appropriate multilayered feed-forward type neural structure, the fractional problem of such initial value is changed into a course of non-linear minimization equations, to some extent. Put differently, interest was sparked in structuring an optimized iterative first-order algorithm to estimate solutions for the origin fractional problem. On top of that, some computer simulation models exemplify the preciseness and well-functioning of the indicated iterative technique. The outstanding accomplished numerical outcomes conveniently reflect the productivity and competency of artificial neural network methods compared to customary approaches.

On the Refinement of the Method of Reducing a System of Linear Differential Equations to a Single Higher-order Equation, Which Makes it Possible to Find a General Solution to the Original

On the Refinement of the Method of Reducing a System of Linear Differential Equations to a Single Higher-order Equation, Which Makes it Possible to Find a General Solution to the Original

Sequences of trees and higher-order renormalization group equations

We define a notion of higher-order renormalization group equations and investigate when a sequence of trees satisfies such an equation. In the strongest sense, the sequence of trees satisfies a kth-order renormalization group equation when applying any choice of Feynman rules results in a Green function satisfying a kth-order renormalization group equation, and we characterize all such sequences of trees. We also make some comments on sequences of trees, which require special choices of Feynman rules in order to satisfy a higher-order renormalization group equation.

The Mechanization thought and its Characteristics Contained in Newton's Mathematics

Understanding the characteristics of Newtonian mathematics is of great value. By analyzing some of Newton’s representative mathematical contents, we found that: Newton used the thought of mathematical mechanization in his mathematics. Newton applied the thought of mathematical mechanization to the study not only of calculus but of arithmetic, algebra, polynomials, higher-order equation, geometry, series, differential equation, etc. Newton’s application of this thought in traditional mathematics is very similar to that of ancient Chinese mathematicians. While in modern mathematics, his procedures are briefer, faster, and more comprehensive than that of other mathematicians at that time. Newton’s wide application of the thought of mathematical mechanization strongly promoted the development of Western mathematics. Like Descartes and Leibniz, Newton was also a pioneer of the development and practical application of the thought of mathematical mechanization in the Western world.

On the dynamics of a higher-order fuzzy difference equation with rational terms

On the dynamics of a higher-order fuzzy difference equation with rational terms

Exact Solutions for Some Singular Linear Ordinary Differential Equations of High Orders via NB1 Polynomials

In this study, we numerically solve the singular linear ordinary differential equations (SLODEs) of higher order using the collocation method based on the NB1 polynomial. An operational matrix form of the given ordinary differential equations (ODEs), the relations of various solutions and the derivatives are obtained from NB1 polynomials. The proposed method reduces the given problem to a linear algebraic equation system, which removes the singularity of ordinary differential equations. The inverse matrix method is used to solve the resulting system to obtain the coefficients of NB1 polynomials. The obtained exact solutions to different problems of high orders show the reliability and accuracy of the presented method.

Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation

ABSTRACT Let d ∈ {1, 2, 3, . . .} and Ω ⊂ ℝ d be open bounded with Lipschitz boundary. Consider the reaction-diffusion parabolic problem P { u t | x | 4 + Δ ( | Δ u | m − 2 Δ u ) = k ( t ) | u | p − 1 u ( x , t ) ∈ Ω × ( 0 , T ) , u ( x , t ) = ∂ x j u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , j ∈ { 1 , … , d } , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω , where T > 0, m ∈ [2, ∞), p ∈ (1, ∞) and 0 ≠ u 0 ∈ W 0 2 , m ( Ω ) ∩ L p + 1 ( Ω ) $0 \ne u_0 \in W^{2,m}_0(\Omega) \cap L^{p+1}(\Omega)$ . We investigate the upper and lower bounds on the blow-up time of a weak solution to (P).

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Computational Methods for Solving Higher-Order (1+1) Dimensional Mixed-Difference Integro-Differential Equations with Variable Coefficients

The main purpose of this article is to present a new technique for solving (1+1) mixeddimensional difference integro-differential Equations (2D-MDeIDEs) in position and time with coefficients of variables under mixed conditions. The equations proposed for the solution represent a link between time and delay in position that has not been previously studied. Therefore, the authors used the technique of separation of variables to transform the 2D-MDeIDE into one-dimensional Fredholm difference integro-differential Equations (FDeIDEs), and then using the Bernoulli polynomial method (BPM), we obtained a system of linear algebraic equations (SLAE). The other aspect of the technique of separation of variables is explicitly obtaining the necessary and appropriate time function to obtain the best numerical results. Some numerical experiments are performed to show the simplicity and efficiency of the presented method, and all results are performed by Maple 18.

Pseudoprocesses related to higher-order equations of vibrations of rods

In this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point.

Boundary value problems with rough boundary data

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn–Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in Lp(R+n)×Lp(Rn−1).

Higher-order complex cubic quintic Ginzburg-Landau equation: Chirped solitary waves

Propagation characteristics of the chirped dissipative solitary waves are investigated within the framework of higher-order complex cubic quintic Ginzburg-Landau equation. A potentially rich set of exact chirped dissipative pulses, such as, bright, dark, grey, antidark, kink, antikink is derived in the presence of the self-steepening, self-frequency shift and nonlinear gain/loss. The linear stability results are corroborated by the direct numerical simulations. The effect of the variation of model parameters on physical quantities, like the speed, amplitude and chirping, is explored.

Higher-Order Nabla Difference Equations of Arbitrary Order with Forcing, Positive and Negative Terms: Non-Oscillatory Solutions

This work provides new adequate conditions for difference equations with forcing, positive and negative terms to have non-oscillatory solutions. A few mathematical inequalities and the properties of discrete fractional calculus serve as the fundamental foundation to our approach. To help establish the main results, an analogous representation for the main equation, called a Volterra-type summation equation, is constructed. Two numerical examples are provided to demonstrate the validity of the theoretical findings; no earlier publications have been able to comment on their solutions’ non-oscillatory behavior.

Localized wave solutions of a higher-order short pulse equation

Localized wave solutions of a higher-order short pulse equation