Solutions Of Differential Equations Research Articles

Solving Fuzzy Delay Differential Equations using Fuzzy Elzaki Transform and Fuzzy Elzaki Decomposition Method

Objective: In this work, the fuzzy Elzaki transform and fuzzy Elzaki decomposition method are used to solve fuzzy delay differential equations. The fuzzy Elzaki decomposition method is a combination of the fuzzy Elzaki transform and Adomian decomposition method in a fuzzy environment. The use of transforms in solving differential equations makes the solution process simpler. Methodology: The fuzzy Elzaki transform is applied in the fuzzy delay differential equation. The nonlinear terms are decomposed using Adomian polynomials in a nonlinear fuzzy delay differential equation. In the transformed domain, the resulting algebraic equation is solved. The solution in the original time domain is obtained by applying the inverse Elzaki transform. Findings: The fuzzy Elzaki transform and fuzzy Elzaki decomposition method are useful for handling the complexities associated with fuzzy functions and delays in differential equations. The technique used to solve the fuzzy delay differential equation gives results with better accuracy compared to the exact solution. The applicability of the method is illustrated with numerical examples. Novelty: The Elzaki transform is a helpful tool for solving delay differential equations with discontinuities. For the resilient solution of complicated differential equations, especially those including nonlinearity and fuzzy logic, the fuzzy Elzaki decomposition approach is useful. It gives an organized way to find the solution to fuzzy delay differential equation by utilizing the advantages of both the Elzaki transform and the Adomian decomposition method in a fuzzy situation. Keywords: Triangular Fuzzy Number, Fuzzy Elzaki Transform, Adomian decomposition method, Fuzzy Elzaki decomposition method, Fuzzy Delay Differential Equations (FDDE)

Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Analytical solutions of the Caudrey-Dodd-Gibbon equation using Khater II and variational iteration methods.

This study focuses on solving the Caudrey-Dodd-Gibbon ( ) equation using the Khater II ( ) method and the Variational Iteration ( ) method. The equation is a pivotal mathematical model in nonlinear wave dynamics, essential for understanding the evolution, interaction, and preservation of wave forms in dispersive media. Its applications span various fields, including fluid dynamics, nonlinear optics, and plasma physics, where it plays a crucial role in analyzing solitons and complex wave interactions. In this research, we meticulously implement the and methods to derive solutions for this nonlinear partial differential equation. Our findings reveal new aspects of the equation's behavior, offering deeper insights into nonlinear wave phenomena. The significance of this study lies in its contribution to advancing the understanding of these phenomena and their practical applications in the academic realm. The results underscore the effectiveness of the employed methods, their innovative contributions, and their relevance to applied mathematics.

Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Delay

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ1,μ2)-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ1,μ2)-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Multidimensional Quantum Generative Modeling by Quantum Hartley Transform

AbstractAn approach for building quantum models based on the exponentially growing orthonormal basis of Hartley kernel functions is developed. First, a differentiable Hartley feature map parameterized by real‐valued argument that enables quantum models suitable for solving stochastic differential equations and regression problems is designed. Unlike the naturally complex Fourier encoding, the proposed Hartley feature map circuit leads to quantum states with real‐valued amplitudes, introducing an inductive bias and natural regularization. Next, a quantum Hartley transform circuit is proposed as a map between computational and Hartley basis. The developed paradigm is applied to generative modeling from solutions of stochastic differential equations, and utilize the quantum Hartley transform for fine sampling from parameterized distributions through an extended register. Finally, the capability of multivariate quantum generative modeling is demonstrated for both correlated and uncorrelated distributions. As a result, the developed quantum Hartley‐based generative models (QHGMs) offer a distinct quantum approach to generative AI at increasing scale.

On entire solutions of certain partial differential equations

On entire solutions of certain partial differential equations

A Finite Difference-Based Adams-Type Approach for Numerical Solution of Nonlinear Fractional Differential Equations: A Fractional Lotka–Volterra Model as a Case Study

Abstract In this paper, we developed an efficient Adams-type predictor–corrector (PC) approach for the numerical solution of fractional differential equations (FDEs) with a power law kernel. The main idea of the proposed approach is to use a linear approximation to the nonlinear problem and then implement finite difference approximations of derivatives. Numerical comparisons with the fractional Adams method are made and simulation results are demonstrated to evaluate the approximation error of the proposed approach. The efficiency of this approach has been depicted by presenting numerical solutions of some test fractional calculus models. Numerical simulation of a fractional Lotka–Volterra model is provided, as a case study, using the proposed approach. The advantage of the proposed approach lies in its flexibility in providing approximate numerical solutions with high accuracy.

Solving Partial Differential Equations Using Fourier Series

This research paper aims to study the methods of solving partial differential equations using Fourier series. Partial differential equations are a powerful and practical tool in mathematics and physics to describe many phenomena and processes that involve continuous change at the level of equations and functions. The basic concepts related to the Fourier series and how to represent functions using it in the field of mathematics will be reviewed. Application examples of partial differential equations, such as waves, heat and diffusion, and how to use the Fourier series to solve them will also be presented. The steps of solving the partial differential equation using the Fourier series will be explained. The importance of Fourier series in the theory of partial differential equations is that periodic functions f(x) defined on (-∞,∞) or functions defined on a finite interval can be represented by an infinite interval of sines and cosines. In this research, the solution of the partial differential equation using the Fourier series was studied with the solution of an example.

Solution of Partial Differential Equations and Their Application by Three Parameters of Integral Transform

Solution of Partial Differential Equations and Their Application by Three Parameters of Integral Transform

Geometric neural operators (gnps) for data-driven deep learning in non-euclidean settings

Abstract We introduce Geometric Neural Operators (GNPs) for data-driven deep learning of geometric features for tasks in non-euclidean settings. We present a formulation for accounting for geometric contributions along with practical neural network architectures and factorizations for training. We then demonstrate how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures of surfaces, (ii) to approximate solutions of geometric partial differential equations on manifolds, and (iii) to solve Bayesian inverse problems for identifying manifold shapes. These results show a few ways GNPs can be used for incorporating the roles of geometry in the data-driven learning of operators.

On the Global Existence of Solution and Lyapunov Asymptotic Practical Stability for Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the asymptotic practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was also established that the main system is asymptotically practically stable in the sense of Lyapunov.

On the Solutions of Nonlinear Implicit ω-Caputo Fractional Order Ordinary Differential Equations with Two-Point Fractional Derivatives and Integral Boundary Conditions in Banach Algebra

This article delves into the analysis of nonlinear implicit ω-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives and integral boundary conditions within the context of Banach algebra. The primary focus is on demonstrating the existence and uniqueness of solutions for these complex fractional differential equations by utilizing Banach's and Krasnoselskii's fixed point theorems. Furthermore, the study explores the stability of these solutions through the Ulam-Hyers and Ulam-Hyers-Rassias stability criteria, thereby assessing the robustness of the proposed model. To illustrate the versatility of the generalized model, several special cases are examined, showcasing its ability to encompass various classical models. The practical applicability of the theoretical findings is underscored through a numerical example, which demonstrates the feasibility and relevance of the proposed methodology. This thorough investigation advances the comprehension of nonlinear fractional differential equations with integral boundary conditions, highlighting the intricate relationship between fractional derivatives, nonlinearities, and integral terms. The results offer significant insights into the behavior and stability of solutions within this demanding mathematical framework.

Periodic solutions of differential equations with periodic constraints

We study the harmonic solutions of periodically perturbed differential equations subjected to a possibly time‐dependent constraint. We obtain a degree theoretic condition ensuring a “global branching” result for the nontrivial periodic solutions. The argument stems from a combination of techniques from the theory of topological degree and differential‐algebraic equations. As an application, the set of harmonic solutions of periodically perturbed implicit differential equations is investigated.

The Improved tan (\(\frac{\Phi(\xi)}{2}\)) -Expansion Method for The Non-dissipative Double Dispersive Equation in Microstructured Solids

In microstructured solids, the non-dissipative double dispersive equation is a fourth-order non-linear partial differential equation that arises in the study of non-dissipative strain wave propagation. Seeking the exact solution of a nonlinear partial differential equations with a physical background is helpful to understand the motion law of matter and to explain the corresponding physical phenomena scientifically. This equation has been studied widely, and there have been a lot of research methods to find exact solutions to this equation. In this manuscript, we try to study the non-dissipative double dispersive equation by the improved tan (\(\frac{\Phi(\xi)}{2}\)) -expansion method. As far as our known, no one has used this method to study the non-dissipative double dispersive equation, partly because of the complicated calculation. We obtain a lot of exact traveling wave solutions, including hyperbolic function solutions, trigonometric function solutions, exponential function solutions, and rational function solutions. Compared with the other methods, some solutions obtained in this manuscript are consistent with existing solutions, which shows the effectiveness of the improved tan (\(\frac{\Phi(\xi)}{2}\)) -expansion method. Through this method, Our manuscript obtains more general and new exact solutions. These solutions may play an important role in engineering and physics. What's more, we plot the 3D graphs of some of the solutions obtained in this manuscript, which helps us understand the physical phenomenon of the non-dissipative double dispersive equation. In the future, we will continue to explore its physical significance from the new analytical solution we have obtained.

Numerical Technique Based on Operational Matrices of Fractional Integration Using ψ$$ \psi $$‐Shifted Chebyshev Polynomials

ABSTRACTIn this paper, we present a numerical scheme based on a modified form of shifted Chebyshev polynomials to find the numerical solution of a class of fractional differential equations. For this purpose, we work out operational matrices of fractional integration of ‐shifted Chebyshev polynomials obtained from shifted Chebyshev polynomials. Finally, the solution to the problem under consideration is obtained by solving a system of algebraic equations that results from the use of operational matrices of integration. The analysis of integer and non‐integer order differential equations is presented to show the convergence of the solution of fractional order differential equation to the corresponding solution of the integer order differential equation. At the end, we present some linear and non‐linear examples to validate the theoretical analysis. Non‐linear examples are solved using Quasilinearization and proposed numerical technique.

Enhancing high-resolution reconstruction of flow fields using physics-informed diffusion model with probability flow sampling

The application of artificial intelligence (AI) technology in fluid dynamics is becoming increasingly prevalent, particularly in accelerating the solution of partial differential equations and predicting complex flow fields. Researchers have extensively explored deep learning algorithms for flow field super-resolution reconstruction. However, purely data-driven deep learning models in this domain face numerous challenges. These include susceptibility to variations in data distribution during model training and a lack of physical and mathematical interpretability in the predictions. These issues significantly impact the effectiveness of the models in practical applications, especially when input data exhibit irregular distributions and noise. In recent years, the rapid development of generative artificial intelligence and physics-informed deep learning algorithms has created significant opportunities for complex physical simulations. This paper proposes a novel approach that combines diffusion models with physical constraint information. By integrating physical equation constraints into the training process of diffusion models, this method achieves high-fidelity flow field reconstruction from low-resolution inputs. Thus, it not only leverages the advantages of diffusion models but also enhances the interpretability of the models. Experimental results demonstrate that, compared to traditional methods, our approach excels in generating high-resolution flow fields with enhanced detail and physical consistency. This advancement provides new insights into developing more accurate and generalized flow field reconstruction models.

Finite orbits of the braid group actions

We study the finite orbits of the braid group Bn action on the space of n×n upper-triangular matrices with 1's along the diagonal. On one hand, we give a necessary condition for a matrix M to be in a finite orbit; on the other hand, we classify and provide lengths of finite orbits in low-dimensional matrices and some other important cases. As the finite orbits on 3×3 matrix were crucial to finding the algebraic solutions of the sixth Painlevé equation, we hope the finite orbits on generic n×n matrices to be useful to finding solutions of higher order Painleve type differential equations.

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

Step-by-step time discrete Physics-Informed Neural Networks with application to a sustainability PDE model

The use of Artificial Neural Networks (ANNs) has spread massively in several research fields. Among the various applications, ANNs have been exploited for the solution of Partial Differential Equations (PDEs). In this context, the so-called Physics-Informed Neural Networks (PINNs) are considered, i.e. neural networks generally constructed in such a way as to compute a continuous approximation in time and space of the exact solution of a PDE.In this manuscript, we propose a new step-by-step approach that allows to define PINNs capable of providing numerical solutions of PDEs that are discrete in time and continuous in space. This is done by establishing connections between the network outputs and the numerical approximations computed by a classical one-stage method for stiff Initial Value Problems (IVPs). Links are also highlighted between the step-by-step PINNs derived here, and the time discrete models based on Runge–Kutta (RK) methods proposed so far in literature. To evaluate the efficiency of the new approach, we build such PINNs to solve a nonlinear diffusion-reaction PDE model describing the process of production of renewable energy through dye-sensitized solar cells. The numerical experiments show that not only the new step-by-step PINNs are able to well reproduce the model solution, but also highlight that the proposed approach can constitute an improvement over existing continuous and time discrete models.