Solutions Of Differential Equations Research Articles

Boundedness of pseudo-differential operators via coupled fractional Fourier transform

In this article, we obtained some fruitful results of the coupled fractional Fourier transform and its kernel. We defined pseudo-differential operators related to coupled fractional Fourier transform on Schwartz spaces and it is shown that their composition is again a pseudo-differential operator. It made further discussion on the composition of two pseudo-differential operators on Sobolev spaces and derived certain norm inequalities on it. Further, we have successfully applied some of the results of the coupled fractional Fourier transform to investigate the solution of nth-order linear non-homogeneous partial differential equation and wave equation. Lastly, we present some examples involving graphs and tables to illustrate the validity of our theoretical findings.

On the Qualitative Analysis of Solutions of Two Fractional Order Fractional Differential Equations

In applied sciences, besides the importance of obtaining the analytical solutions of differential equations with constant coefficients, the qualitative analysis of the solutions of such equations is also very important. Due to this importance, in this study, a qualitative analysis of the solutions of a delayed and constant coefficient fractal differential equation with more than one fractional derivative was performed. In the equation under consideration, the derivatives are the Riemann–Liouville fractional derivatives. In the proof of the obtained results, Laplace transform formulas of the Riemann–Liouville fractional derivative and some inequalities are used. We also provide some examples to check the accuracy of our results.

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Stochastic theta methods for random periodic solution of stochastic differential equations under non-globally Lipschitz conditions

Stochastic theta methods for random periodic solution of stochastic differential equations under non-globally Lipschitz conditions

Solvability of a boundary value problem with integrable conditions for partial differential equations of fractional order

Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. The study of existence and uniqueness, periodicity, asymptotic behavior, stability, and methods of analytic and numerical solutions of fractional differential equations have been studied extensively in a large cycle works; Especially, The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, we only mention here the apriori estimate method (energy inequality method). The current piece of work concentrates on exploring the existence and uniqueness of a solution for a non-linear boundary value problem that has integrable conditions in the case of fractional partial differential equations. For this we split the proof into two sections: linear and non-linear problem; for the associated linear problem, we derive the a priori bound and demonstrate the density of the operator generated by the problem posed; we solve the non-linear problem by introducing a iterative process. The results show the efficiency of energy inequality method in the case of time fractional order differential equations with integrable conditions our results illustrate the existence and uniqueness of the continuous dependence of solution on fractional order. The work of the authors could be considered as a contribution to the development of the functional analysis method. And to conclude we argue that the research carried out in this article could serve as a contribution for possible studies in the field of applied mathematics.

Symbolic computation in algebra, geometry, and differential equations

In this survey article we describe how symbolic computation in algebra and geometry leads to symbolic, i.e., formula solutions of algebraic differential equations. Symbolic solutions of algebraic differential equations can be derived from parametrizations of corresponding algebraic varieties. Such parametrizations in turn can be computed by elimination methods, i.e., methods for solving systems of polynomial equations.

An Intelligent Parameter Identification Algorithm of Linear Fuzzy Fractional Differential Equation

Abstract The traditional linear generation fuzzy fractional differential equation parameter identification algorithm lacks the update of parameter identification process, has large amount of calculation, slow convergence speed of parameter identification and strong dependence on initial values. In this paper, a new intelligent recognition algorithm for linearly generated fuzzy fractional differential equations is proposed. The parameters of the equation are re expressed by the constant level set. The piecewise constant level set algorithm based on equation parameter intelligent identification is used to solve the steady-state solution of fractional differential equation, and the non convergence problem caused by too much calculation is solved. A new algorithm scheme for linearly generating fuzzy fractional differential equations is established, the constraints of the level set of the differential equations are calculated, and the updated algorithm for parameter identification of the equation is obtained. The evolutionary algorithm is used to solve the updating algorithm to realize the intelligent identification algorithm of linear fractional fuzzy differential equations. Experimental results show that the algorithm has the advantages of fast convergence speed, high calculation accuracy and low initial value.

Output feedback finite‐time boundary control for an unstable heat PDE with spatially varying coefficients

AbstractThis article studies the output feedback finite‐time boundary control for unstable heat systems with the spatially varying coefficient. First, a finite‐time observer with switched gains under a state‐dependent switching law is designed in order to estimate the states of the system in a finite‐time only exerting one displacement boundary measurement. Next, an observer‐based linear finite‐time control is planned. Namely, a linear switched control under a state‐dependent switching law is proposed to vanish every solution of an unstable heat partial differential equation with spatially varying coefficients in a finite time. We also present explicit forms for the proposed observer gains and output feedback finite‐time controller. Finally, some numerical simulations are provided to confirm the theoretical results.

Optical soliton solutions of nonlinear differential Boussinesq water wave equation via two analytical techniques

In this article, we examine the optical soliton solutions of the 4th-order nonlinear Boussinesq water wave equation using the modified (G′/G2)− expansion and F-expansion methods. We utilize the similarity transformation to transform the partial differential form of the equation into an ordinary differential form. These techniques present a variety of solutions, including bright, dark, singular, dark-periodic, and singular-periodic soliton solutions. We plot the derived solutions in several profiles, such as 2D, 3D, and contour, to illustrate their physical appearance. The determined findings might be useful and have an enormous effect on the research of nonlinear phenomena in a variety of physical areas of science, like shallow water waves, acoustics, fluid dynamics, laser optics, communication systems, and heat transfer. The achieved results demonstrate the validity, applicability, and effectiveness of the presented method. We plot and validate the found solutions using the computational program MATHEMATICA.

New solutions of the general elliptic equation and its applications to the new (3 + 1)-dimensional integrable Kadomtsev-Petviashvili equation

In this article, we introduce many new Jacobi elliptic function solutions to the general elliptic equation. Consequently, the Jacobi elliptic function expansion method is improved to accommodate the general elliptic equation and its new solutions for constructing exact traveling wave solutions of nonlinear partial differential equations (NLPDEs). Moreover, the improved method is used to obtain new explicit solutions for the (3+1)-dimensional integrable Kadomtsev-Petviashvili (KP) equation. This method can be applied to many other NLPDEs as well for obtaining new exact solutions.

On the ϕ-Order of Growth of Solutions of Complex Linear Differential Equations near an Essential Singular Point

In this article, we will take care about an interesting topic which is the study of the ϕ-order of growth of solutions of some given linear differential equations with analytic coefficients in C-{z0}, where z0∈C represents an essential singularity. What we’ll do in this paper is a generalization of the work of Long and Zeng by introducing the concept of the ϕ–order of growth near an essential singularity.

A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces

A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces

On Asymptotic Properties of Solutions for Differential Equations of Neutral Type

On Asymptotic Properties of Solutions for Differential Equations of Neutral Type

Applications of the Homotopy-based Fourier transform method for the dynamic solutions of differential equations

Applications of the Homotopy-based Fourier transform method for the dynamic solutions of differential equations

Nonlinear Waves in (3 + 1)-Dimensions: Multi-Soliton Solutions of a Partial Differential Equation

Nonlinear Waves in (3 + 1)-Dimensions: Multi-Soliton Solutions of a Partial Differential Equation

Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method

The study of soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) has gained prominence recently because of its ability to realistically recreate complex physical processes. Numerous mathematical techniques have been devised to handle the problem of NFPDEs where soliton solutions are difficult to obtain. Due to their accuracy in reproducing complex physical phenomena, soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) have recently attracted interest. Several mathematical techniques have been devised to tackle the difficult task of solving non-finite partial differential equations (NFPDEs) soliton. Studies of soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) have garnered increased attention recently due to its capacity to accurately represent complex physical processes. Due to the difficulty of obtaining soliton solutions, NFPDEs can be solved using a wide variety of mathematical methods. In this way, it facilitates the extraction of the recently found abundance of optical soliton solutions. To further understanding of the results, the study also includes contour and three-dimensional images that visually depict particular optical soliton solutions for particular parameter selections, suggesting the existence of different soliton structures in the nonlinear fractional Kundu–Eckhaus equation (NFKEE) region. It is shown that the proposed technique is quite powerful and effective in solving several nonlinear FDEs.

Fractional Order and Numerical Solution Application of Modeling in the Management of Examination Misconducts

The fractional order model that represents the spread of Examination misconduct using compartments of the population of honest students (susceptible), those lightly involved in misconduct (exposed), seriously involved ones (infected), and quitters (removed) is provided. The fractional order derivative is considered in the Caputo sense. To determine the epidemic forecast and persistence, we calculate the reproduction number. Analyzing the stability of this scheme ensures a non-negative and unique solution within the defined domain (0,1). Employing the Laplace-Adomian Decomposition Method aids in estimating the solution for the nonlinear fractional differential equations. Utilizing infinite series helps derive solutions for these equations, ensuring convergence to their precise values. The results obtained align with outcomes from the traditional Differential Transformed Method. Finally, numerical results and an outstanding graphic simulation are presented.

Optimal Relaxed Control for a Decoupled G-FBSDE

AbstractIn this paper we study a system of decoupled forward-backward stochastic differential equations driven by a G-Brownian motion (G-FBSDEs) with non-degenerate diffusion. Our objective is to establish the existence of a relaxed optimal control for a non-smooth stochastic optimal control problem. The latter is given in terms of a decoupled G-FBSDE. The cost functional is the solution of the backward stochastic differential equation at the initial time. The key idea to establish existence of a relaxed optimal control is to replace the original control problem by a suitably regularised problem with mollified coefficients, prove the existence of a relaxed control, and then pass to the limit.

Analytic Solution of the Time-Fractional Partial Differential Equation Using a Multi-G-Laplace Transform Method

In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) are proved. The solution of the system of time-fractional partial differential equations is addressed using a new analytical method. This technique is a combination of the multi-G-Laplace transform and decomposition methods (MGLTDM). Moreover, we discuss the convergence of this method. Two examples are provided to check the applicability and efficiency of our technique.

A numerical solver based on Haar wavelet to find the solution of fifth-order differential equations having simple, two-point and two-point integral conditions

A numerical solver based on Haar wavelet to find the solution of fifth-order differential equations having simple, two-point and two-point integral conditions