Nonhomogeneous Equations Research Articles

An optimal control problem for resource utilisation by microorganisms

Decomposition of organic matter controls the flow of carbon and nutrients in terrestrial and aquatic ecosystems. Several kinetic laws have been proposed to describe decomposition rates, but they neglect adaptation of the microbial decomposer to environmental conditions. Here we formalise decomposition as an optimal control problem by assuming that microorganisms regulate the uptake rate of a substrate to maximise their growth over the period of decomposition. The result is an optimal control problem consisting of two differential equations and auxiliary conditions that determine the optimal value of the control variable (the uptake rate), the remaining substrate at any given time, and the optimal completion time. This problem serves as a case study to illustrate the solution of differential equations and optimal control problems for students in undergraduate courses. The mathematical analysis of the problem requires rewriting the differential equations in reverse time along with the solution of a nonhomogeneous linear first order differential equation. We then return to modelling with some biologically motivated questions about how the parameters of the model representing environmental conditions and microbial functional traits affect the outcome. Finally, we discuss alternative ways to use the material with students.

Global Well-Posedness for 2D Nonhomogeneous Magneto-Micropolar Equations with Density-Dependent Viscosity

Global Well-Posedness for 2D Nonhomogeneous Magneto-Micropolar Equations with Density-Dependent Viscosity

Numerical approximation of the Cauchy non-homogeneous time-fractional diffusion-wave equation with Caputo derivative using shifted Chebyshev polynomials

In this paper, we introduce a numerical technique for solving the Cauchy non-homogeneous time-fractional diffusion-wave equation with the Caputo derivative operator. The key idea behind our approach is to employ the roots of shifted Chebyshev polynomials to collocate the problem in the time dimension. By introducing certain functions to homogenize the resulting system of ordinary differential equations, we obtain an approximate solution to the problem. To assess the reliability of our method, we conduct a thorough comparison with existing numerical techniques available in the literature. Through extensive experimentation, we demonstrate the effectiveness and accuracy of our proposed technique. Notably, the results highlight the efficiency and precision of our method in approximating solutions for the considered problem. Furthermore, we investigate the convergence of our method and provide theoretical insights into its performance. Our findings support the validity and robustness of the proposed approach. Overall, this paper contributes a valuable numerical tool for addressing the Cauchy non-homogeneous time-fractional diffusion-wave equation, offering enhanced efficiency and accuracy compared to existing methods.

Energy harvesting induced by the vibration of reciprocating-piston compressor subjected to repetitive impulse

In this study, discrete event time-variant parameter of a lumped SISO dynamical system is obtained for a novel energy harvester. The harvester utilizes the resonance vibration of a coupled fluid-mechanic system, which is comprised of a reciprocating-piston, a tunable oscillator and twin electric turbine generators. The system model is simplified by an underdamped spring-mass-damper with tunable parameters subjected to periodic or arbitrary impulse forces. We developed a method that solves general problems underlining non-homogenous ordinary second order differential equation with discrete event time-variant parameters. The method uses the superposition to compile the general solution by segmenting the problem into ODE equations with constant coefficient. Each segment corresponds to a change in system, and it is solved analytically with recursively updated initial conditions. We utilized this method to tune the system’s natural frequency such that the accumulated energy is maximum within desired time range. We obtained the general solution of the displacement response due to varying parameters and input impulse force. We obtained the displacement peaks and their time of occurrence that could help understand the structural design limitation. We showed system displacement responses (constructive and destructive) corresponding to various impulse repetition patterns. We derived the relationship between energy, volume displacement, and volumetric flow rate (or piston velocity) and provided illustrative case studies. One finding shows that for a given impulse condition the accumulative energy exhibits local maximum and minimum within a narrow band of natural frequency range. The simulations showed the feasibility of generating considerable power from electric turbine when the air flow is driven by a resonating piston.

Non-homogeneous Riemannian gradient equations for sum of squares of Bures–Wasserstein metric

We investigate Riemannian gradient equations on the open convex cone Pm of all m×m positive definite Hermitian matrices, for the function WA(X)=∑j=1nwjdW2(X,Aj)where dW denotes the Bures–Wasserstein metric. On the special case where the gradient of the weighted sum of squares of the Bures–Wasserstein metrics vanishes, its unique solution is known as the Wasserstein mean of A1,…,An. We discuss the existence and uniqueness of solutions for non-homogeneous Riemannian gradient equations at which vector fields are constant and congruence transformations. Furthermore, we establish some bounds for these solutions.

Time-dependent creep analysis of ultra-high-temperature functionally graded rotating disks of variable thickness

Functionally graded materials (FGMs) are high temperature-resistant materials that can simultaneously maintain metallic tenacity and anti-corrosive properties. Nevertheless, using FGMs during a multi-year service life at ultrahigh temperatures is crucial. Hence, the time-dependent creep response of variable-thickness rotating disks made of FGM is investigated. Four different disk profiles of linear, concave, convex, and uniform are considered. The material's creep properties are defined by the Bailey-Norton creep law. Loading is a rotation-based mechanical body force and a uniform temperature throughout the disk. Simultaneous solution of equilibrium, stress-strain, and strain-displacement equations yields a non-homogenous differential equation containing variable and time-dependent coefficients. In an attempt to optimize the computation cost, Bat and Fish algorithms were used to optimize the initial strain presumptions. Semi-analytical solution of this differential equation gives radial and circumferential stress histories and displacement histories. To confirm the solution method, initial thermo-elastic radial stress, and the effective stress history are validated with the existing literature; there is a good agreement between the results. In addition, the finite element software ABAQUS was used to model the FGM disk thermo-elastic behavior, and the result was compared with the semi-analytical solution results. This study emphasizes the significance of accounting for creep effects in the design of FGM rotating disks, as remarkable changes in their displacements and stresses occur over time. This study emphasizes the significance of accounting for creep effects in the design of FGM rotating disks, as notable changes in their displacements and stresses occur over time.

Efficient Finite Difference Approaches for Solving Initial Boundary Value Problems in Helmholtz Partial Differential Equations

This study presents numerical solutions for initial boundary value problems of homogeneous and nonhomogeneous Helmholtz equations using first- and second-order difference schemes. The stability of these methods is rigorously analyzed, ensuring their reliability and convergence for a wide range of problem instances. The proposed schemes’ robustness and applicability are demonstrated through several examples, accompanied by an error analysis table and illustrative graphs that visually represent the accuracy of the solutions obtained. The results confirm the effectiveness and efficiency of the proposed schemes, making them valuable tools for solving Helmholtz equations in practical applications.

Fast Generalized Sliding Sinusoidal Transforms

Discrete cosine and sine transforms closely approximate the Karhunen–Loeve transform for first-order Markov stationary signals with high and low correlation coefficients, respectively. Discrete sinusoidal transforms can be used in data compression, digital filtering, spectral analysis and pattern recognition. Short-time transforms based on discrete sinusoidal transforms are suitable for the adaptive processing and time–frequency analysis of quasi-stationary data. The generalized sliding discrete transform is a type of short-time transform, that is, a fixed-length windowed transform that slides over a signal with an arbitrary integer step. In this paper, eight fast algorithms for calculating various sliding sinusoidal transforms based on a generalized solution of a second-order linear nonhomogeneous difference equation and pruned discrete sine transforms are proposed. The performances of the algorithms in terms of computational complexity and execution time were compared with those of recursive sliding and fast discrete sinusoidal algorithms. The low complexity of the proposed algorithms resulted in significant time savings.

Discrete heat equation with irregular thermal conductivity and tempered distributional data

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$ . We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell ^{2}_{s}(\hbar \mathbb {Z}^n)$ , $s \in \mathbb {R}$ , which is enough to prove the well-posedness in the space of tempered distributions $\mathcal {S}'(\hbar \mathbb {Z}^n)$ . Notably, when $s=0$ , we show that for $\hbar \rightarrow 0$ , the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb {R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb {Z}^n$ .

Method of Variation of Parameters for the Third-Order Linear Proportional Dynamic Equations

The method of variation of parameters for the third-order linear nonhomogeneous differential equations is given in this article, utilizing the proportional derivative and its characteristics on the time scale. Then, an example is provided to illustrate how to apply the provided approach.

On the growth of solutions of complex linear differential equations with analytic coefficients in $\overline{\mathbb{C}}\backslash\{z_{0}\}$ of finite logarithmic order

In this article, we study the growth of solutions of homogeneous and non-homogeneous complex linear differential equations where the coefficients are analytic functions in the extended complex plane except a finite singular point with finite logarithmic order. We extend some previous results obtained very recently by Fettouch and Hamouda.

A sediment transport theory based on distortion-free-boundary nonhomogeneous fluid flows

Over the last 50 years several sediment transport models in coastal environments based on Shallow Water(SW) type models have been developed in the literature. The water flow over an abrupt moving topography quickly spatially variable becomes accelerated and strongly varied arising the turbulence (distortion). The acceleration and strong variation of the flow facilitate the transport of a large quantity of sediments present at the bottom while modifying it. The mathematical models based on SW type models widely used to describe the sediment transport phenomena do not account the distortion effects. Indeed, it is well-known that the SW models are derived from first order approximation of long wave theory. The acceleration and strong variation of the water flow near the bottom is due to the distortion of the horizontal velocity profile along the vertical direction. One can regard distortion as a combination of strain and rotation. The effect of the rotational component is to weaken the effect of the strain somewhat. In this work, we put in place a king theory of sediment transport derived from the second order approximation of long wave theory that can describe sediment transport processes in distortion-free-boundary nonhomogeneous fluid flows. The derived model accounts the distortion (fluctuation with great correlation lengths) that creates the turbulence. Moreover, the model differentiates the fluid velocity from sediment velocity (phase-lag) near the sediment bed. The proposed theory significantly reduces the modeling errors observed in several sediment transport models based on nonhomogeneous shallow water equations and has a great potential to increase the predictive power of sediment transport models in rivers, lakes, coastal flows, ocean basins and so on. The proposed theory improves several existing sediment transport theories recently developed in the literature and can be apply with some degree confidence.

Modeling a building during an earthquake to demonstrate resonance

ABSTRACT Spring-mass systems are presented as an application of second-order, constant coefficient differential equations in many differential equations textbooks. The phenomenon of resonance can then be analysed after nonhomogeneous differential equations are introduced. With some simplifying assumptions, the movement of the roof of a one-story building can be modeled as a spring-mass system and ground vibrations, such as those from an earthquake, can then be viewed as an external force acting on this system. In this paper, I will discuss a project I wrote and use in which students analyse the movement of a building resulting from an earthquake. This requires students to think in terms of amplitude/displacement (rather than position) and observe how this is affected by the frequency of an external force.

Screw Dynamics of a Multibody System by a Schoenflies Manipulator

This paper presents a screw dynamics method for multi-rigid-body systems. It establishes the relationship between velocity screw (twist) and force screw (wrench) in the theorem of momentum screw. Then, the structure of a 2UPS + 2UPU parallel manipulator is introduced as an example of application. By analyzing the constraint wrench of each limb and the twist of the moving platform in the theorem of momentum screw, the dynamics equation for solving the driving force makes the method very convenient for computer programming. The dynamics equation can be reorganized into a non-homogeneous linear equation and establishes the relationship between the constraint wrench of each limb and the twist of the moving platform in screw coordinate. The outstanding advantage of the algorithm proposed in this paper is that the average calculation time is only 83.21% of that of the Newton–Euler method for the 2UPS + 2UPU parallel manipulator on the same computer. This methodology provides a convenient canonical form for the dynamics analysis of multi-rigid-body systems.

Cα regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

Cα regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

SVPWM Strategy With Minimum Common-Mode Voltage for Multilevel Converter Combining the Concept of the Nearest Level

Multilevel converters are widely used in medium- and high-voltage high-power industrial applications. Space vector pulse width modulation (SVPWM) is one of the very promising modulation strategies for multilevel converters. However, the SVPWM strategy is difficult to implement as the number of levels increases. The output common-mode voltage (CMV) of the converter is another problem in the implementation of SVPWM strategy. In this paper, a new SVPWM strategy with minimum CMV for multilevel converter is proposed by combining the concept of the nearest level. The proposed SVPWM strategy only utilizes simple arithmetic operations and logical judgments, and completely avoids trigonometric functions and irrational number operations. Compared with the existing SVPWM methods, the proposed SVPWM strategy can quickly calculate the switching state with the minimum CMV in real-time by the concept of the nearest level, and avoids solving the non-homogeneous linear equations in the process of calculating the switching state corresponding to the space vector. It can be extended to any level converter without additional computational burden. Finally, the proposed SVPWM strategy is verified with extensive simulation and experimental studies.

Existence and regularity of strong solutions to a nonhomogeneous Kelvin-Voigt-Cahn-Hilliard system

A well-known diffuse interface model consists of the Navier-Stokes equations nonlinearly coupled with the convective Cahn-Hilliard equation. This system describes the evolution of an incompressible isothermal Newtonian mixture of binary fluids. In this article we investigate a variant of this model, which consists of the nonhomogeneous Kelvin-Voigt equations coupled with the Cahn-Hilliard equations. We prove the existence of global weak and strong solutions in a two and three dimensions. Furthermore, we prove some regularity results for the strong solutions and show their uniqueness in both two or three-dimensional bounded domains. Lastly, we also solve a problem of uniqueness of regular solutions that was left open by Giorgini and Temam (2020) [5].

On a first order linear singular differential equation in the space K’

We propose in this work to describe all the generalized-function solutions of the non-homogeneous first-order linear singular differential equation with A, B two real numbers, s and p∈ N, n ≥ 1， q∈Z+, in the space of generalized functions K’. In the case of a second right-hand side consisting of an s-order derivative of the Dirac-delta function, we have completely investigated the considered equation when we look for the solution in the form of y(x)=∑k=0NCkδ(k)(x), with the unknown coefficients Ck which we have determined case by case, taking into account the relationship between the parameters inside. On the basis of what has been done, we focus our present research on applying the principle of superposition of the solutions that is conducting us to the awaited result when we also maintain the classical solutions of the homogeneous equation which remains the same.

On the computation of rational solutions of linear integro-differential equations with polynomial coefficients

We develop the first algorithm for computing rational solutions of scalar integro-differential equations with polynomial coefficients. It starts by finding the possible poles of a rational solution. Then, bounding the order of each pole and solving an algebraic linear system, we compute the singular part of rational solutions at each possible pole. Finally, using partial fraction decomposition, the polynomial part of rational solutions is obtained by computing polynomial solutions of a non-homogeneous scalar integro-differential equation with a polynomial right-hand side. The paper is illustrated by examples where the computations are done with our Maple implementation.

Iterative learning algorithms for boundary tracing of fractional diffusion equations

AbstractWe study P‐type and PI ‐type iterative learning control (ILC) schemes for boundary tracing problem of nonhomogeneous fractional diffusion equations. Based on Sobolev imbedding theorem, we derive sufficient conditions for the convergence of four ILC schemes in the sense of ‐norm. Numerical examples are presented to illustrate effectiveness of the proposed control methods. The results show that closed‐loop ILC scheme converges faster than open‐loop ILC scheme; moreover, PI ‐type ILC scheme outperforms P‐type and PI‐type ILC schemes in terms of the convergence speed.