Solutions Of Differential Equations Research Articles

Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

Until now, little investigation has been done to examine the existence and uniqueness of solutions for fractional differential equations on star graphs. In the published articles on the subject, the authors used a star graph with one junction node that has edges with the other nodes, although there are no edges between them. These graph structures do not cover more generic non-star graph structures; they are specific examples. The purpose of this study is to prove the existence and uniqueness of solutions to a new family of fractional boundary value problems on the tetramethylbutane graph that have more than one junction node after presenting a labeling mechanism for graph vertices. The chemical compound tetramethylbutane has a highly symmetrical structure, due to which it has a very high melting point and a short liquid range; in fact, it is the smallest saturated acyclic hydrocarbon that appears as a solid at a room temperature of 25 °C. With vertices designated by 0 or 1, we propose a fractional-order differential equation on each edge of tetramethylbutane graph. Employing the fixed-point theorems of Schaefer and Banach, we demonstrate the existence and uniqueness of solutions for the suggested fractional differential equation satisfying the integral boundary conditions. In addition, we examine the stability of the system. Lastly, we present examples that illustrate our findings.

(N,λ)-periodic solutions to abstract difference equations of convolution type

This work primarily focuses on (N,λ)-periodic sequences and their applications. To begin, we provide a brief overview of (N,λ)-periodic sequences and introduce several results. Secondly, in terms of applications and main objective, we establish sufficient criteria for both the existence and uniqueness of (N,λ)-periodic mild solutions for abstract difference equations of convolution type. Furthermore, we present illustrative examples to highlight our key findings.

On the numerical solution of second order delay differential equations via a novel approach

On the numerical solution of second order delay differential equations via a novel approach

Companion-based multi-level finite element method for computing multiple solutions of nonlinear differential equations

The utilization of nonlinear differential equations has resulted in remarkable progress across various scientific domains, including physics, biology, ecology, and quantum mechanics. Nonetheless, obtaining multiple solutions for nonlinear differential equations can pose considerable challenges, particularly when it is difficult to find suitable initial guesses. To address this issue, we propose a pioneering approach known as the Companion-Based Multilevel Finite Element Method (CBMFEM). This novel technique efficiently and accurately generates multiple initial guesses for solving nonlinear elliptic semi-linear equations containing polynomial nonlinear terms through the use of finite element methods with conforming elements. As a theoretical foundation of CBMFEM, we present an appropriate and new concept of the isolated solution to the nonlinear elliptic equations with multiple solutions. The newly introduced concept is used to establish the inf-sup condition for the linearized equation around the isolated solution. Furthermore, it is crucially used to derive a theoretical error analysis of finite element methods for nonlinear elliptic equations with multiple solutions. A number of numerical results obtained using CBMFEM are then presented and compared with a traditional method. These not only show the CBMFEM's superiority, but also support our theoretical analysis. Additionally, these results showcase the effectiveness and potential of our proposed method in tackling the challenges associated with multiple solutions in nonlinear differential equations with different types of boundary conditions.

Asymptotic Behavior of <i>n</i>th Order Impulsive Differential Equations

This paper devotes to the asymptotic behavior of all solutions of nth order impulsive differential equations. Based on impulsive differential inequality, boundedness and zero tendency of every solution for nth order impulsive differential equations are obtained. In addition, we derive globally uniformly exponential stability of every solution under Lyapunov function and impulsivetechnique, and these results are extend to <i>n</i>th order differential equations with periodic coefficient and periodic impulse. Meanwhile, an example with simulations are provided to verify the conclusion.

Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications

Abstract In this manuscript, we prove new extensions of Nashine, Wardowski, Feng-Liu, and Ćirić-type contractive inequalities using orbitally lower semi-continuous functions in an orbitally complete b b -metric space. We accomplish new multivalued common fixed point results for two families of dominated set-valued mappings in an ordered complete orbitally b b -metric space. Some new definitions and illustrative examples are given to validate our new results. To show the novelty of our results, applications are given to obtain the solution of nonlinear integral and fractional differential equations. Our results expand the hypothetical consequences of Nashine et al. (Feng–Liu-type fixed point result in orbital b-metric spaces and application to fractal integral equation, Nonlinear Anal. Model. Control. 26 (2021), no. 3, 522–533) and Rasham et al. (Common fixed point results for new Ciric-type rational multivalued-contraction with an application, J. Fixed Point Theory Appl. 20 (2018), no. 1, Paper No. 45).

Application of Hayman’s Theorem to Directional Differential Equations With Analytic Solutions in the Unit Ball

In this paper, we investigate analytic solutions of higher order linear non-homogeneous directional differential equations whose coefficients are analytic functions in the unit ball. We use methods of theory of analytic functions in the unit ball having bounded L-index. Our proofs are based on application of inequalities from analog of Hayman’s theorem for analytic functions in the unit ball. There are presented growth estimates of their solutions which contain parameters depending on the coefficients of the equations. Also, we obtained sufficient conditions that every analytic solution of the equation has bounded L-index in the direction. The deduced results are also new in one-dimensional case, i.e. for functions analytic in the unit disc. Keywords: Analytic function, analytic solution, slice function, unit ball, directional differential equation, growth estimate, bounded L-index in direction.

Bounded solutions for semilinear differential equations

Bounded solutions for semilinear differential equations

Koopman neural operator as a mesh-free solver of non-linear partial differential equations

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-Bénard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modeling, and precipitation forecasting).

Solving a class of Thomas–Fermi equations: A new solution concept based on physics-informed machine learning

This paper presents a novel physics-informed machine learning approach, designed to approximate solutions to a specific category of Thomas–Fermi differential equations. To tackle the inherent intricacies of solving Thomas–Fermi equations, we employ the quasi-linearization technique, which transforms the original non-linear problem into a series of linear differential equations. Our approach utilizes collocation least-squares support vector regression, leveraging fractional Chebyshev functions for finite interval simulations and fractional rational Chebyshev functions for semi-infinite intervals, enabling precise solutions for linear differential equations across varied spatial domains. These selections facilitate efficient simulations across both finite and semi-infinite intervals. Key contributions of our approach encompass its versatility, demonstrated through successful approximation of solutions for diverse Thomas–Fermi problem types, including those featuring non-local integral terms, Bohr radius boundary conditions, and isolated neutral atom boundary conditions defined on semi-infinite domains. Furthermore, our method exhibits computational efficiency, surpassing classical collocation methods by solving a sequence of positive definite linear equations or quadratic programming problems. Notably, our approach showcases precision, as evidenced by experiments, including the attainment of the initial slope of the renowned Thomas–Fermi equation with an impressive 39-digit precision.

On the Boundedness of Solutions of a Non-autonomous Differential Equation of Second Order

We study the boundedness of the solutions to a non-autonomous differential equation of second order. With this work, we improve some stability results in the literature of boundedness of the solutions. We give two examples to illustrate the theoretical analysis in this work. 2000 Mathematics Subject Classification. 34C11, 34D20

Solving integral and differential equations via fixed point results involving F-contractions

In this article, we aim to establish fixed point results within the framework of an orthogonal complete metric space by employing an F-weak contraction. Our research extends and generalizes several well-established results found in the existing literature. To substantiate the validity of our findings, we have included illustrative examples. Additionally, our discoveries empower us to ascertain both the existence and uniqueness of solutions for both differential and integral equations.

Convergence of Fibonacci–Ishikawa iteration procedure for monotone asymptotically nonexpansive mappings

In uniformly convex Banach spaces, we study within this research Fibonacci–Ishikawa iteration for monotone asymptotically nonexpansive mappings. In addition to demonstrating strong convergence, we establish weak convergence result of the Fibonacci–Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure, we approximate the solution of the Caputo-type nonlinear fractional differential equation.

Numerical Solution of the Linear Fractional Delay Differential Equation Using Gauss–Hermite Quadrature

Fractional order differential equations often possess inherent symmetries that play a crucial role in governing their dynamics in a variety of scientific fields. In this work, we consider numerical solutions for fractional-order linear delay differential equations. The numerical solution is obtained via the Laplace transform technique. The quadrature approximation of the Bromwich integral provides the foundation for several commonly employed strategies for inverting the Laplace transform. The key factor for quadrature approximation is the contour deformation, and numerous contours have been proposed. However, the highly convergent trapezoidal rule has always been the most common quadrature rule. In this work, the Gauss–Hermite quadrature rule is used as a substitute for the trapezoidal rule. Plotting figures of absolute error and comparing results to other methods from the literature illustrate how effectively the suggested approach works. Functional analysis was used to examine the existence of the solution and the Ulam–Hyers (UH) stability of the considered equation.

Numerical simulations for fractional differential equations of higher order and a wright-type transformation

In this work, a new relationship is established between the solutions of higher order fractional differential equations and a Wright-type transformation. Solutions could be interpreted as expected values of functions in a random time process. As applications, we solve the fractional beam equation, fractional electric circuits with special functions as external sources, derive d’Alembert’s formula and show the existence of explicit solutions for a general fractional wave equation with variable coefficients. Due to this relationship, we present two methods for simulating solutions of fractional differential equations. The two approaches use the interpretation of the Caputo derivative of a function as a Wright-type transformation of the higher derivative of the function. In the first approach, we use the Runge–Kutta method of hybrid orders 4 and 5 to solve ordinary differential equations combined with the Monte Carlo integration to conduct the Wright-type transformation. The second method uses a feedforward neural network to simulate the fractional differential equation.

Exact solutions of stochastic Burgers–Korteweg de Vries type equation with variable coefficients

We will present exact solutions for three variations of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are time-dependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV–Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equations.

Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

This paper introduces gradient-based adaptive neural networks to solve local fractional elliptic partial differential equations. The impact of physics-informed neural networks helps to approximate elliptic partial differential equations governed by the physical process. The proposed technique employs learning the behaviour of complex systems based on input-output data, and automatic differentiation ensures accurate computation of gradient. The method computes the singularity-embedded local fractional partial derivative model on a Hausdorff metric, which otherwise halts the computation by available approximating numerical methods. This is possible because the new network is capable of updating the weight associated with loss terms depending on the solution domain and requirement of solution behaviour. The semi-positive definite character of the neural tangent kernel achieves the convergence of gradient-based adaptive neural networks. The importance of hyperparameters, namely the number of neurons and the learning rate, is shown by considering a stationary anomalous diffusion-convection model on a rectangular domain. The proposed method showcases the network’s ability to approximate solutions of various local fractional elliptic partial differential equations with varying fractal parameters.

Tracking optimal feedback control under uncertain parameters

Optimal control problems of tracking type for a class of linear systems with uncertain parameters in the dynamics are investigated. An affine tracking feedback control input is obtained by considering the minimization of an energy-like functional depending on a finite ensemble of training/sample parameters. It is computed from the solution of an associated differential Riccati equation. Simulations are presented showing the tracking performance of the computed input for trained as well as untrained parameters.

An Efficient Algorithm for Basic Elementary Matrix Functions with Specified Accuracy and Application

If the matrix function f(At) posses the properties of f(At)=gf(tkA, then the recurrence formula fi−1=gfi,i=N,N−1,⋯,1,f(tA)=f0, can be established. Here, fN=f(AN)=∑j=0majANj,AN=tkNA. This provides an algorithm for computing the matrix function f(At). By specifying the calculation accuracy p, a method is presented to determine m and N in a way that minimizes the time of the above algorithm, thus providing a fast algorithm for f(At). It is important to note that m only depends on the calculation accuracy p and is independent of the matrix A and t. Therefore, f(AN) has a fixed calculation format that is easily computed. On the other hand, N depends not only on A, but also on t. This provides a means to select t such that N is equal to 0, a property of significance. In summary, the algorithm proposed in this article enables users to establish a desired level of accuracy and then utilize it to select the appropriate values for m and N to minimize computation time. This approach ensures that both accuracy and efficiency are addressed concurrently. We develop a general algorithm, then apply it to the exponential, trigonometric, and logarithmic matrix functions, and compare the performance with that of the internal system functions of Mathematica and Pade approximation. In the last section, an example is provided to illustrate the rapid computation of numerical solutions for linear differential equations.

Application of Fixed Point Theorems to Differential Equation in b-Multiplicative Metric Spaces

Objectives: In this manuscript, some fixed point theorems have been provided in b-multiplicative metric spaces. Methods: We proved the unique fixed point theorems using the Banach contraction principle and generalized Lipschitz contractive mappings. Findings: We established the P Property and the T-Stability of Picard’s iteration in b-multiplicative metric spaces. We also provide an example to demonstrate the result. Novelty: Using our results, we obtained the existence and uniqueness of solutions for ordinary multiplicative differential equations with initial value problems. 2020 Mathematics Subject Classification: 47H10, 54H25. Keywords: Fixed Point, Generalized Lipschitz Contractive, Picard’s iteration, b-Multiplicative Metric Space (b − MMS), T -Stability, Differential equation