Nonhomogeneous Linear Differential Equations Research Articles

An algorithm for solving linear nonhomogeneous quaternion-valued differential equations and some open problems

<p style='text-indent:20px;'>Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper.</p>

Differential polynomials generated by solutions of second order non-homogeneous linear differential equations

Differential polynomials generated by solutions of second order non-homogeneous linear differential equations

Mittag-Leffler-Hyers-Ulam Stability of a linear differential equations of second order using Laplace transform

In this paper, we investigate the Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam-Rassias stability of a homogeneous and non-homogeneous linear differential equations of second order by using Laplace transforms.

Submergence depth modeling of oil well reservoirs and applications

In oilfield production, it is important to be able to acquire the reservoir status of oil wells in real time, especially for low or ultra-low permeability oil wells. In this paper, based on the reservoir percolation characteristics of the oil layer and combined with the oil well permeability, liquid pressure and oil well pressure formulas, the non-homogeneous linear differential equation for oil well pressure is derived, based on which a submergence depth model of oil well reservoirs is established. The parameter merging is applied to reduce the parameter dimension of the submergence depth model; this approach substantially reduces the parameter dimension and application complexity and effectively improves the model application range. The model not only conforms to the reservoir percolation law of the oil layer but also applies to the balance laws of other substances with potential energy balancing ability. The application methods studied (namely, the extraction of the greatest common divisor, the weighted dichotomy and least-squares curve fitting) successively reduce the calculation error and expand the application scope. The least-squares curve fitting method causes the submergence depth error to reach the minimum of the 2-norm, and the number of iterations can be accurately assessed by the convergence speed of halving the numerical calculation error in every iteration. This method is suitable not only for the zero submergence depth case in the initial state but also for non-zero submergence depth cases. The experimental results show that the proposed submergence depth model for oil well reservoirs accurately reflects the change laws of submergence depth and time. Combined with the proposed application methods, the corresponding submergence depth can be obtained at any time point. The model and application methods provide strong support for formulating scientific oil production plans and implementing reasonable production methods for oil wells.

Analytical solutions of one-line model of shoreline change on the evolution of river delta on a coast bounded by solid boundaries

A river delta is a landform created when rivers empty their water and sediments into another body of water, such as an ocean, lake, or another river. This study proposes a new analytical solution for the one-line model of shoreline change by solving the nonhomogeneous linear ordinary partial differential equation. It describes river delta formation from the river sediment supply (point source) on a bounded coast, where the ends of the coast are fixed by solid boundaries. Furthermore, new analytical solutions are obtained by applying the property of linear superposition. These solutions describe the retreat of the river delta shoreline near the river mouth caused by the reduction of river sediment supply. They also outline the river delta shoreline recovery following the recovery of river sediment supply. A new analytical solution that describes shoreline evolution for an instantaneously shifting river mouth on the bounded coast is proposed by employing the same method. In addition, the resultant data is compared with the corresponding evolutions of the river delta shoreline on the coast without solid boundaries. Theoretical results indicate that river delta formation on the bounded coast is more rapid than on an unbounded coast. This can be attributed to the restriction of sediment loss by the solid boundaries. The boundary influence time of the solid boundary is defined as the time elapsed when solid boundaries interrupt the sediment transport and result in a significant difference of shoreline evolution compared to the coast without the solid boundaries. Its value in dimensionless form varies from 0 to 0.07, depending on the location of the river mouth, which acts as a point source. During the period of sediment reduction, the extent of retreat and the region of retreat near the river mouth are smaller and narrower than the case without solid boundaries. The deposition occurs on the side where the location of the river mouth channel is advancing, while erosion is observed on the other side. The results of this study were not verified with measured data. However, the derived solution would be a benchmark for numerical solutions and beneficial to the primary design stage, educational purposes, river delta, and river management authorities.

Solving Non-Homogeneous Linear Ordinary Differential Equations Using Memristor-Capacitor Circuit

Inhomogeneous linear ordinary differential equations (ODEs) and systems of ODEs can be solved in a variety of ways. However, hardware circuits that can perform the efficient analog computation to solve them are rarely in the literature. To address such problems, this paper proposes a general method of using a memristor-capacitor (M-C) circuit to solve inhomogeneous linear ODEs and systems of ODEs of any order in initial value problems. The M-C circuit can match the coefficients of the equations sought by adjusting the memristor resistance value according to the coefficient formula proposed in the paper, which has higher programmability. Then, some ODEs and systems of ODEs are given in the paper as examples to evaluate the proposed method. According to the comparison results based on MATLAB software simulation and the simulation based on OrCAD software, the designed M-C circuit has an effective improvement in speed and the accuracy exceeds 99.95% in software simulation. Based on practical verification, this paper gives the actual M-C circuit experiment based on PCB. Moreover, the proposed method can be used to quickly solve the object motion state in the spring mass damping system in actual engineering, and the accuracy can reach 99.98%.

The growth of solutions of non-homogeneous linear differential equations

In this paper, we have considered second order non-homogeneous linear differential equations whose coefficients are entire functions. We have established conditions ensuring the non-existence of finite order solution of such type of differential equations. Moreover, we have also extended our results to higher order non-homogeneous differential equations.

Some Properties on The [p,q]-Order of Meromorphic Solutions of Homogeneous and Non-homogeneous Linear Differential Equations With Meromorphic Coefficients

In the present paper, we investigate the $\left[p,q\right] $-order of solutions of higher order linear differential equations \begin{equation*} A_{k}\left(z\right) f^{\left( k\right) }+A_{k-1}\left( z\right) f^{\left(k-1\right)}+\cdots +A_{1}\left( z\right) f^{\prime }+A_{0}\left( z\right) f=0 \end{equation*} and \begin{equation*} A_{k}\left( z\right) f^{\left( k\right) }+A_{k-1}\left( z\right) f^{\left(k-1\right) }+\cdots +A_{1}\left( z\right) f^{\prime }+A_{0}\left( z\right) f=F\left( z\right), \end{equation*} where $A_{0}\left( z\right) ,$ $A_{1}\left( z\right) ,...,A_{k}\left(z\right) \not\equiv 0$ and $F\left( z\right) \not\equiv 0$ are meromorphic functions of finite $\left[ p,q\right] $-order. We improve and extend some results of the authors by using the concept $\left[ p,q\right] $-order.

Experimental and theoretical assessment of dual-slope single-basin solar still with the circular cross-sectional hollow-fins

The experimental and theoretical assessment of the dual-slope single-basin solar still has been presented with and without circular cross-section hollow-fins appended on the absorber plate. Two alike dual-slope single-basin solar stills were constructed, one with hollow-fins and the other without hollow-fins. The 10 mm depth of brackish water was maintained in the basin. For the three consecutive days in the month of April, experiments were conducted under the meteorological situations of Valsad (20.61°N, 72.91°E). Using the basic principles of heat transfer and extended surface, dedicated theoretical models have been formulated for the still with and without hollow-fins. To study the transient behaviour of the system, differential equations of various components of solar still developed in theoretical models were solved using the analytical method used to solve the nonhomogeneous linear first-order differential equation. The theoretical results have been compared with experimental measurements. The absorber plate area for absorption of solar irradiation was increased by 23.78% with the inclusion of hollow-fins. With an enhancement of 47.35%, an average yield of 1285.2 ml/m2 was achieved for the still with hollow-fins. The average daily efficiency of 17.93% was achieved with an enhancement of 51.74% for the still with hollow-fins.

Representation of solutions of nonhomogeneous conformable fractional delay differential equations

• Linear and nonlinear time-delay equation with conformable fractional derivative is proposed. • Conformable delay perturbation of matrix exponential function is defined for the first time. • Solution of the equation is determined with the help of Conformable delay perturbation of matrix exponential function. • Its existence, uniqueness and stability are proved via Banach Fixed Point Theorem and Ulam-Hyers Stability. This paper is about the conformable fractional delay equations. We offer a conformable delay perturbation of matrix exponential function to give the representation of solutions for linear nonhomogeneous conformable fractional delay differential equations . Lastly, the existence and uniqueness of solutions and Ulam-Hyers stability of the equations are proved.

Solution of One-dimensional Partial Differential Equation with Higher-Order Derivative by Double Laplace Transform Method

Many problems in natural and engineering sciences such as heat transfer, elasticity, quantum mechanics, water flow, and others are modelled mathematically by partial differential equations. Some of these problems may be linear, nonlinear, homogeneous, non-homogeneous, and order greater or equal one. Finding the theoretical solution to these problems with less cumbersome techniques is an active area of research in the aforementioned field. In this research paper, we have developed a new application of the double Laplace transform method to solve homogeneous and non-homogeneous linear partial differential equations (pdes) with higher-order derivatives (i.e order n where n≥2) in science and engineering. We discussed a brief theory of double Laplace transforms that helped in its application. The main advantage of our method is the reduction of computational effort in finding solution to pdes. Another major benefit of our method is solving problems in the form of (21) directly by transforming to an algebraic equation where the inverse double Laplace transform is implemented for analytical solution, unlike other integral transform methods that would first transform to a system of ODEs before they are solved, is it also very effective in solving linear high-order partial differential equations and yield fast convergence. We present a well-simplified solution for easier comprehension by upcoming researchers.

Multi-delayed perturbation of Mittag-Leffler type matrix functions

In this paper, we introduce a multivariate determining function and propose a multi-delayed perturbation of Mittag-Leffler type matrix function. It is an extension of the classical Mittag-Leffler type matrix function and delayed Mittag-Leffler type matrix function. With the help of the multi-delayed perturbation of Mittag-Leffler type matrix function, we give an explicit analytical solution to linear nonhomogeneous Riemann-Liouville fractional multi-delay differential equations of order l−1<α≤l. Our results are new even for the case 1<α<2 and for the classical (α=2) second order single delay and multi-delay linear (fractional) differential equations.

Instability of second-order nonhomogeneous linear difference equations with real-valued coefficients

In J. Comput. Anal. Appl. (2020), pp. 152--165, the author dealt with Hyers--Ulam stability of the second-order linear difference equation $\Delta_h^2x(t)+\alpha \Delta_hx(t)+\beta x(t) = f(t)$ on $h\mathbb{Z}$, where $\Delta_hx(t) = (x(t+h)-x(t))/h$ and $h\mathbb{Z} = \{hk|\,k\in\mathbb{Z}\}$ for the step size $h&gt;0$; $\alpha$ and $\beta$ are real numbers; $f(t)$ is a real-valued function on $h\mathbb{Z}$. The purpose of this paper is to clarify that the second-order linear difference equation has no Hyers--Ulam stability when the step size $h&gt;0$ and the coefficients $\alpha$ and $\beta$ satisfy suitable conditions. Finally, a necessary and sufficient condition for Hyers--Ulam stability is obtained.

Probabilistic analysis of a class of impulsive linear random differential equations via density functions

An important class of non-homogeneous first-order linear random differential equations subject to an infinite sequence of square impulses with random intensity is studied. In applications, these equations are useful to model the dynamics of a population with periodic harvesting and migration under uncertainties. The solution is explicitly obtained via the first probability density function assuming an arbitrary joint density for all model parameters. Probabilistic stability analysis is carried out through the densities of the random sequences of minima and maxima. All the theoretical results are fully illustrated through two numerical examples.

Mathematical Modelling in Engineering with Integral Transforms via Modified Adomian Decomposition Method

In this work three integral transforms through modified Adomian decomposition method (ADM) are proposed to obtain the approximate analytical solution of different types of mathematical models arising in physical problems. These transformations are applied for both homogeneous and non-homogeneous linear differential equations. The efficiency and accuracy of the proposed methods are implemented through higher order non-homogeneous ordinary differential equations. Numerical tests are reported for applicability of the current scheme based on different transformations and compared with exact solutions.

Solving systems of conformable linear differential equations via the conformable exponential matrix

In this research paper, we discuss systems of conformable linear differential equations. The conformable fundamental exponential matrix has been used to express the solution of the homogeneous and nonhomogeneous systems. The method of variation of parameters has been investigated to find the particular solution of the conformable system of nonhomogeneous linear differential equations. Several illustrative examples have been provided at the end of our study to validate all obtained results.

Explicit solutions of non-homogeneous difference equations from finite potent endomorphisms

The aim of this work is to show the consistency of all systems of non-homogeneous linear difference equations of the form , where is a finite potent endomorphism of an arbitrary vector space V and . An algorithm to compute the set of solutions of these systems is given. In particular, the method offered is valid for computing the explicit solutions of the system of non-homogeneous difference equations , with A being a finite square matrix.

Growth of solutions of non-homogeneous linear differential equations and its applications

In this paper, we investigate the growth properties of solutions of the non-homogeneous linear complex differential equation $L\left(f\right)=b\left(z\right)f+c\left(z\right)$, where $L\left(f\right)$ is a linear differential polynomial and $b\left( z\right) $, $c\left( z\right) $ are entire functions and give some of its applications on sharing value problems.

A Mathematical Model for the Effects of Physical Activity and Nutrition in Patients with Diabetes Mellitus

The disease of diabetes is ravaging the world’s population affecting the lives of many people mainly the economically active groups. Its effects result in a decrease of the most productive world’s labor force and consequently reducing the Gross Domestic Product (GDP). The objective of this study was to develop a mathematical model for Blood Glucose Regulatory System (BGRS) using Linear Nonhomogeneous Differential Equations. This model incorporated nutrition and physical activity, to be able to manage Diabetes mellitus in such a way that, the peak, which is the time period for insulin to be most effective in lowering blood sugar, is within the acceptable range. It is worth noting that after doing physical activity for some minutes, we have that the glucose and insulin levels in the blood plasma of diabetic patient decreases back to their normal levels .This means that if a patient observes the correct diet and does physical activity regularly and mostly once per day for a certain period, there is no need of going for any insulin supplement. This is more beneficial because the body may become resistant to introduction of insulin with time.

COMPARATIVE STUDY OF SOLUTION METHODS OF NON-HOMOGENEOUS LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT

The methods to obtain a solution of non-homogeneous linear ordinary differential equations with constant coefficients vary from type of output function. The present paper addresses the undetermined coefficients, variations of parameters methods and discussed the rewards and drawbacks of these methods with examples. The method of undetermined coefficients is derived from the variation of parameters when the out function is an exponential function.