Nonhomogeneous Equations Research Articles

Interpretation of exact solution for fuzzy fractional non-homogeneous differential equation under the Riemann–Liouville sense and its application on the inventory management control problem

This article is presented for manifestation of the non-homogeneous linear fractional differential equation under fuzzy uncertainty. Taking the initial values and the coefficients of the fractional differential equations to be fuzzy numbers, the solutions are categorized into different problems and sub-problems. Based on the three different combinations of the fuzzy initial values and fuzzy coefficients, the theoretical foundation in this article is primarily divided into three major problems. Then, each problem again contains two different sub-problems according to the sign of the coefficients. Finally, the solutions for each of the sub-problems are given in the sense of two different cases of $${}^{\text{RL}}[\left(i\right)-\alpha ]$$ and $${}^{\text{RL}}[\left(ii\right)-\alpha ]$$ differentiability. The strong and the weak solution criteria have been discussed here. Also, the existence and uniqueness criterion for solution of the initial-valued fuzzy fractional differential equation has been established in this article. Finally, an EPQ model of deteriorated items is discussed for its production phase only as a proper validation of proposed theory. The greatness of the consideration of fuzzy fractional differential equation over crisp integer differential equation, fuzzy integer differential equation and crisp fractional differential equation to describe the EPQ model is established in this current article. In this context, a new defuzzification technique is developed to compare the fuzzy and crisp phenomena related to the EPQ model.

Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.

Strong solutions to the 2D Cauchy problem of nonhomogeneous magnetohydrodynamic equations with vacuum

In this paper, we study the strong solutions to the Cauchy problem of nonhomogeneous incompressible magnetohydrodynamic (MHD) equations on the whole two-dimensional (2D) space with vacuum as far field density. In particular, the initial density can have compact support. If both the initial density and the initial magnetic field decay not too slow at infinity, the 2D Cauchy problem of the nonhomogeneous incompressible MHD equations admits a unique local strong solution.

Reducing the near boundary errors of nonhomogeneous heat equations by boundary consistent methods

Abstract In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.

The analysis of factors influencing the trajectory of the vibrating screen

Due to the fact that the shape of the trajectory of the sieve vibrating screen significantly affects the efficiency of cleaning the drilling mud from the drilled rock, this article identifies and analyzes the factors influencing the formation of the trajectory of the sieve. The study of the nature of the motion of the particles of the drilled rock is carried out. The motion-equation of the point of the jigging frame, respectively, and a certain section of the screen can be described by an non-homogeneous second-order differential equation (Newton equation). The equation of the trajectory of the jigging frame is also considered. The influence of the amount of drilling mud, which is on the vibrating screen at a certain time-point, on the trajectory is taken into account in this equation. The drilling mud complex system consisting of liquid and solid fractions – sludge, sand and sludge – is taken into account. In this case, the fluid and clay move without separation from the sieve, and sludge and coagulated clay – with separation from the transport surface. Depending on the position on the sieve, each component of the drilling mud at different time periods moves with different trajectories and different other variable parameters along the length of the sieve. Therefore, the main purpose of the research is to obtain a graphical real trajectory of the selected point of the jigging frame of vibrating screens, to compare it with theoretical graphical dependences and to establish parameters that affect the change in the amplitude of jogging frame vibrations in real operating conditions. The coverage of the problem and the main results of the research are performed in real conditions (when vibrating screens work on the drilling rigs). The research is performed using simple recording devices installed at six places – on the left and right sides of the jigging frame above the front and rear shock absorbing springs, and in the center of stuff – between springs. All the real trajectories of certain points of the jigging frame and screen can be described using certain Lissajous figures. However, it was found that not all trajectories coincide completely with the classic Lissajous figures, because the trajectory is influenced by external factors that cause chaotic motion of the jigging frame – the Sommerfeld effect, as well as non-uniformity of drilling fluid delivery. The nature of the movement of the vibrating plane - the screen - is also affected by the rigidity of the vibrating support, including the entire structure of the vibrating screens. The phenomenon of resonance is possible in the area of ​​vibration supports. In addition, it was previously established that the trajectory of the point of a certain area of ​​the screen is a spatial figure that requires additional theoretical and experimental studies. When moving the drilling fluid on the sieve with the toss, the cleaning efficiency will be much higher due to the increase in vibration force. Based on the analysis of the results of the factors influencing the trajectory of the sieve, preliminary recommendations are given for the correct adjustment of the vibrating screen.

Elzaki Substitution Method for Solving Nonlinear Partial Differential Equations with Mixed Partial Derivatives Using Adomian Polynomial

In this paper we apply a new method, named Elzaki Substitution Method to solve nonlinear homogeneous and nonhomogeneous partial differential equations with mixed partial derivatives, which is based on Elzaki Transform. The proposed method introduces also Adomian polynomials and the nonlinear terms can be handled by the use of this polynomials. The proposed method worked perfectly to find the exact solutions of partial equations with mixed partial derivatives without any need of linearization or discretization in comparison with other methods such as Method of Separation of Variables (MSV) and Variation Iteration Method (VIM). Some illustrative examples are given to demonstrate the applicability and efficiency of proposed method.

On the existence of solutions to an inhomogeneous pantograph type equation with singular coefficients

We establish the existence of solutions to an initial boundary value problem that involves a certain class of nonhomogeneous functional partial differential equations of the pantograph type with singular time dependent coefficients. The problem is motivated from a cell division equation. The techniques for solving such problems are rare due to the nonlocal term and the singularity in coefficients.

Global well-posedness and decay estimates of strong solutions to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection

We deal with an initial boundary value problem of nonhomogeneous Boussinesq equations for magnetohydrodynamics convection in two-dimensional domains. We prove that there is a unique global strong solution. Moreover, we show that the temperature converges exponentially to zero in H1 as time goes to infinity. In particular, the initial data can be arbitrarily large and vacuum is allowed. Our analysis relies on energy method and a lemma of Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997).

The Mean First Passage Time of the Stochastic Resonance Driven by Dual-Sequence-Frequency-Hopping Signal and Noise

The mean first passage time (MFPT) represents the dynamic characteristic of stochastic resonance (SR). The study focuses on how can the Dual-Sequence-Frequency-Hopping (DSFH) signal influence the MFPT and any difficulty in solving the MFPT problem considering the DSFH signal. In this current study, the SR system driven by DSFH signal and Gaussian white noise is described with the parameters of the signal amplitude, the frequency of Intermediate Frequency (IF) of the receptive DSFH signal, the SR system parameter, scale transformation coefficient, the noise intensity, and the sampling multiple, firstly. Secondly, under the assumption that MFPT is small aqueous about the domain of 0, the nonautonomous differential equation with MFPT is transformed to a nonhomogeneous differential equation with one unknown variable coefficient of second order. Finally, the numerical solution of MFPT can be obtained by the method of Runge–Kutta. Theoretical and simulation results are shown as below: (1) the effect of the signal amplitude, the IF frequency, the noise intensity, the SR system parameter, and the scale transformation coefficient, for decrease the MFPT, are positive; however, the effect of the sampling multiple is negative; (2) the MFPT cannot follow the dynamic period of the SR controlled by the IF frequency, when SNR is low; (3) when SNR = −12 dB, the sampling multiple is 200, the IF frequency is 2100, and the duty cycle reaches 25% (available for DSFH signal detection with peek or valley decision Liu et al. (2019)), so we need to decrease the IF frequency or increase the SNR for further availability.

Robust Boundary Vibration Control of Uncertain Flexible Robot Manipulator with Spatiotemporally-varying Disturbance and Boundary Disturbance

In this paper the vibration control problem is addressed for the Euler-Bernoulli beam with system parameters uncertainties, spatiotemporally-varying disturbance, and boundary disturbance. By using global sliding-mode boundary control (GSMBC) through method of lines (MOL), a robust boundary control design is suggested to diminish the perturbations of uncertain Euler-Bernoulli beam and to compensate the influence of the spatiotemporally-varying disturbance and boundary disturbance. Dynamics of the Euler-Bernoulli beam are described by non-homogenous hyperbolic partial differential equation (PDE) and ordinary differential equations (ODEs). MOL is employed to acquire the beam dynamics represented by ODE system in lieu of PDE system, therefore a precise solution is obtained by solving the resulting ODE system. Then, GSMBC is established for mitigating the vibrations of the beam affected by system parameters uncertainties, spatiotemporally-varying disturbance, and boundary disturbance. Chattering phenomena is avoided by using exponential reaching law supported by a relay function. Exponential convergence and stability robustness of the closed-loop system are assured by Lyapunov direct approach. In the end, simulation outcomes show that the GSMBC-based MOL scheme is valid for vanishing the vibrations of the uncertain Euler-Bernoulli beam efficiently.

APPROXIMATE SOLUTION OF AN INITIAL-BOUNDARY VALUE PROBLEM FOR AN NONHOMOGENEOUS SECOND-ORDER DIFFERENTIAL EQUATION OF MIXED TYPE WITH TWO DEGENERATE LINES

This work is devoted to the study of an approximate solution of the initial-boundary value problem for the second order mixed type nonhomogeneous differential equation with two degenerate lines. Similar equations have many different applications, for example, boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in magneto hydrodynamics, in the theory of infinitesimal bindings of surfaces, in the theory of shells, in predicting the groundwater level, in plasma modeling, and in mathematical biology. In this paper, based on the idea of A.N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an apriori estimate of the solution of the equation, we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. The results of the numerical solutions and the approximate solutions of the original problem were presented in the form of tables. The regularization parameter is determined from the minimum estimate of the norm of the difference between exact and approximate solutions.

The fractional Landweber method for identifying the space source term problem for time-space fractional diffusion equation

This paper is devoted to solve an inverse problem for identifying the source term of a time-fractional nonhomogeneous diffusion equation with a fractional Laplacian in a non-local boundary. Based on the expression of the solution for the direct problem, the inverse problem for searching the space source term is converted into solving the first kind of Fredholm integral equation. The conditional stability for the inverse source problem is investigated. The fractional Landweber method is used to deal with this inverse problem and the regularized solution is also obtained. Furthermore, the convergence rates for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Several numerical examples are given to show the proposed method is efficient and stable.

On Solvability in the Sense of Sequences for some Non-Fredholm Operators with Drift and Anomalous Diffusion

We study the solvability of certain linear nonhomogeneous elliptic equations and establish that, under some technical assumptions, the L2-convergence of the right-hand sides yields the existence and convergence of solutions in an appropriate Sobolev space. The problems involve differential operators with or without Fredholm property, in particular, the one-dimensional negative Laplacian in a fractional power, on the whole real line or on a finite interval with periodic boundary conditions. We prove that the presence of the transport term in these equations provides regularization of the solutions.

Local Existence and Uniqueness of Strong Solutions to the Two Dimensional Nonhomogeneous Incompressible Primitive Equations

In this article, we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions. The initial vacuum is allowed.

Exact Solutions of Generalized Riemann Problem for Nonhomogeneous Shallow Water Equations

In this paper, we consider quasilinear hyperbolic system of balance laws describing one-dimensional nonhomogeneous shallow water equations with generalized Riemann initial data. We obtain exact solutions to the shallow water equations with friction by using differential constraint method. A special case of the obtained solution provides well known rarefaction wave to the homogeneous case of the governing equations. We construct a convenient example for the generalized Riemann problem and study the behavior of the solution profiles.

Large deviations and Berry–Esseen inequalities for estimators in nonhomogeneous diffusion driven by fractional Brownian motion

Abstract We are interested in bounds on the large deviations probability and Berry–Esseen type inequalities for maximum likelihood estimator and Bayes estimator of the parameter appearing linearly in the drift of nonhomogeneous stochastic differential equation driven by fractional Brownian motion.

The Flat Galactic Rotation Curves from First Principles: A Novel Logarithmic Potential Originating in Laplace’s Equation

Abstract This note reports flat galactic rotation curves coinciding with observation and originating in Laplace’s classical field equation. Dark matter, identified with a primordial fluid populating the entire universe, is represented by the homogeneous Klein–Gordon equation (KGE). The key element is the novel solution for the KGE discovered by the leading author in 1994, expressed as a permanent background  -potential plus an entangled  -potential where time and distance appear as a ratio. The  -potential may be viewed as solution to a non-homogeneous Laplace equation, for which the general solution contains in addition to Newton’s 1/r potential a new logarithmic Zwicky potential overlooked thus far. The ensuing 1/r-Zwicky force predicts the observed flat galatic rotation curves exhibiting the observed limiting speed at long distance.

Mellin Transform and Integro-Differential Equations with Logarithmic Singularity in the Kernel

In the one-dimensional domain it is studied the unique solvability of a class of integro-differential equations with logarithmic singularity in the kernels. The solution of the considering integro-differential equations with logarithmic singularity in the kernels on dependent of roots of corresponding characteristic equations is obtained in an explicit form. The first part of paper deals with the introduction of the Mellin transformation as a holomorphic function in the complex plane. The definition of this integral transform is given and its main properties are described in details. The second part deals with the application of the Mellin transform to the solution of one-dimensional integro-differential equations with the logarithmic singularity in the kernels. Here, in order to obtain particular solution of nonhomogeneous equations, we applied the analogies of Duhamel’s theorem. It is shown that, if the kernel of studied equation has first order logarithmic singularity of polynomial’s form, then to this equation corresponds characteristic (algebraic) equation of the third order. In this case the solution is found by means of degree of logarithmic and trigonometric functions. If the integro-differential equation has $$n$$ -th order logarithmic singularity of polynomial’s form, then to this equation corresponds $$n$$ -th order characteristic equation. At three main cases the solution is found in an explicit form. Also, it is established that the degree of logarithmic singularity acts to number of linearly independent solutions of given classes of integro-differential equations.

Forced oscillations of the main gas pipeline open section during the cleaning piston passage

The problem of forced oscillations of an open section of a gas pipeline during the cleaning piston passage belongs to the type of problems of forced oscillations of one-dimensional elastic objects under the influence of a moving inertial load on them. Currently, there are two ways to solve such problems. The first way is related to the integration of the partial differential equation and the solution of such problems is a superposition of eigen-oscillations and accompanying oscillations. The second way does not involve the integration of the partial dif-ferential equation. Methods of generalized coordinates, generalized displacements and various numerical methods belong to the second type of solving. None of the mentioned methods is simple. Therefore, the authors suggest the method, in which the first mathematical model provides the determination of forced oscillations of the gas pipeline section during the passage of the cleaning piston without taking into account its inertial load on the gas pipeline. In future, on the basis of the first model it is planned to develop the second mathematical model which will provide an approximate determination of the deflections of the pipeline axis, taking into account the inertial load of the piston on the pipeline. The purpose of this article is to obtain a solution to the problem of forced oscillations of the pipeline section during the passage of the cleaning piston without taking into account the inertial forces on the pipeline. The problem is solved by partial differential equation, Fourier method is applied. The right side of the non-homogeneous differential equation is decomposed into an infinite series, which is the sum of the produc-tions of the eigenfunctions of the pipeline section free oscillations and the unknown function of time. After finding out this function, the authors determine the unknown time function in the Fourier method and hence the solution of the problem in the form of an infinite series, the summands of which lessen rapidly. The authors calculate the deflections of the pipeline axis along the entire section of the gas pipeline for different points of time, as well as deflections of individual sections changing in time and moments of deflection.

Exact solutions and Hyers–Ulam stability for fractional oscillation equations with pure delay

This paper presents the exact solutions of the homogeneous and the non-homogeneous fractional delay oscillation equations with order α∈(1,2). For non-homogeneous fractional delay oscillation equation, the Laplace transform is taken up as solving method in that it efficiently avoids some restrictions to the non-homogeneous term. Thereafter, we investigate the Hyers–Ulam stability of the fractional delay oscillation equation.