Solutions Of Differential Equations Research Articles

Stochastic stability of solutions for a fourth-order stochastic differential equation with constant delay

In this paper, we present sufficient conditions to ensure the stochastic asymptotic stability of the zero solution for a specific type of fourth-order stochastic differential equation (SDE) with constant delay. By reducing the fourth-order SDE to a system of first-order SDEs, we utilize a fourth-order quadratic function to derive an appropriate Lyapunov functional. This functional is then employed to establish standard criteria for the nonlinear functions present in the SDE. The stability result obtained in this study is novel and extends the existing findings on stability in fourth-order differential equations. Additionally, we provide an illustrative example to demonstrate the significance and accuracy of our main result.

Iterative Construction of Fixed Points for Functional Equations and Fractional Differential Equations

This paper proposes some iterative constructions of fixed points for showing the existence and uniqueness of solutions for functional equations and fractional differential equations (FDEs) in the framework of CAT (0) spaces. Our new approach is based on the M ∗ -iterative scheme and the class of mappings with the KSC condition. We first obtain some ∆ and strong convergence theorems using M ∗ -iterative scheme. Using one of our main results, we solve a FDE from a broad class of fractional calculus. Eventually, we support our main results with a numerical example. A comparative numerical experiment shows that the M ∗ -iterative scheme produces high accurate numerical results corresponding to the other schemes in the literature. Our results are new and generalize several comparable results in fixed point theory and applications.

Variable substitution methods in the solution of stochastic ordinary differential equations and their applications

Abstract In this paper, we first propose numerical solution methods for stochastic ordinary differential equations by using the two-step Maruyama method and Euler-Maruyama method in variable substitution, and analyze the mean-square compatibility, mean-square convergence and mean-square linear stability of the corresponding numerical methods, respectively. Finally, 10,000 times value experiments are conducted to verify the convergence accuracy and stability of the variable substitution methods. The results show that the Euler method simulates this equation when taking steps h = 0.05, β = 5 and σ = 2 for numerical experiments, and the numerical results obtained are convergent but unstable. On the other hand, the Euler-Maruyama method with variable substitution is consistent with the real solution. It shows that variable substitution is an important method for solving stochastic ordinary differential equations with high convergence and stability, which is of great significance for the solution of stochastic ordinary differential equations.

Analytical study of nonlinear models using a modified Schrödinger’s equation and logarithmic transformation

Optical solitons are special waves that maintain their shape while traveling. Hence, solitary optical waves have great significance when it comes to representing pulse propagation through nonlinear partial differential equations. In our research, we have uncovered unique solutions for the formation of solitary wave patterns in the equation of the Heisenberg ferromagnetic spin chain (HFSC). This equation aptly describes the behavior of electromagnetic waves in contemporary magnetism. By employing innovative techniques based on logarithmic transformations and analytical methods, we have obtained various solution forms expressed in a concise manner using elementary functions. We have tested the correctness of these solutions by substituting them directly into the main equation. These recent discoveries provide novel perspectives on the complex realm of solitons as depicted by this model. Moreover, since the model finds applications in fiber optic communications, fluid dynamics, and other fields, the implications of these discoveries are broad. The utilized methods stand out for being simple, reliable, and capable of creating fresh solutions for nonlinear partial differential equations in the realm of mathematical physics. The research findings presented here showcase the effectiveness of the applied methods in reliably studying nonlinear phenomena in the HFSC equation and other nonlinear problems in mathematical physics. By utilizing these tools, scholars have the opportunity to expand and enhance their understanding of the complex mathematical frameworks underlying real-world problems.

Lévy Langevin Monte Carlo

Analogously to the well-known Langevin Monte Carlo method, in this article we provide a method to sample from a target distribution π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\pi }$$\\end{document} by simulating a solution of a stochastic differential equation. Hereby, the stochastic differential equation is driven by a general Lévy process which—unlike the case of Langevin Monte Carlo—allows for non-smooth targets. Our method will be fully explored in the particular setting of target distributions supported on the half-line (0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0,\\infty )$$\\end{document} and a compound Poisson driving noise. Several illustrative examples conclude the article.

Extension of optimal auxiliary function method to non-linear fifth order lax and Swada-Kotera problem

In this article, a semi-analytical approach known as the optimal auxiliary function method is extended to the approximate solution of non-linear partial differential equations. The fifth order lax and swada-kotera equations are taken as test examples. Utilizing the well-known least squares method, the optimal convergence control parameter values in the auxiliary function have been determined. The outcomes of the proposed method are contrasted with those of a new iterative approach and a homotopy perturbation method. It has been demonstrated that the suggested method for solving non-linear partial differential equations is straightforward and rapidly convergent. The numerical outcomes demonstrate the effectiveness and reliability of the suggested approach. Additionally, using higher order approximations can increase the suggested method's accuracy.

Twisted Hypersurfaces in Euclidean 5-Space

The twisted hypersurfaces x with the (0,0,0,0,1) rotating axis in five-dimensional Euclidean space E5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In E5, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces x are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of x are determined. Additionally, the Laplace–Beltrami operator relation of x is given.

Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations

In this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solutions of nonlinear partial differential equations. When estimating the diffusion coefficient, it was necessary to construct an estimate of the second derivative based on inaccurate observations of the solution of the diffusion equation. This procedure required consideration of two reference points when determining the first and second partial derivatives of the solution of the diffusion equation. To analyse one-soliton solutions of partial differential equations, a series of techniques have been developed that allow one to estimate the parameters of the solution itself, but not its equation. These techniques are used to estimate the parameters of one-soliton solutions of the equations kdv, mkdv, Sine–Gordon, Burgers and nonlinear Schrodinger. All the considered estimates were tested during computational experiments.

Unpredictable solutions of compartmental quasilinear differential equations

n the present research, quasilinear differential equations with compartmental periodic unpre- dictable coefficients and perturbations are investigated. The compartmental periodic unpredictable functions present a new type of recurrency. The problem of existence and uniqueness of unpredictable solutions is stud- ied for the systems. Numerical simulations which illustrate chaotic and periodic behavior in outputs of the dynamics are provided

Дополнительные граничные условия в задачах теплопроводности с неоднородными граничными условиями

Using additional boundary conditions (ABC) and an additional sought function (ASF), a solution to the heat conduction problem with non-homogeneous boundary conditions has been obtained. ABC allows satisfying the differential equation at the boundaries, leading to its fulfillment both inside the domain and without resorting to integration over Cartesian coordinates. ASF reduces the original partial differential equation to a temporal ordinary equation, from which the eigenvalues of the boundary value problem are determined, as formulated in the classical methods of the Sturm-Liouville problem, stated in spatial variables. Thus, this work considers an alternative way of determining eigenvalues. The integration constants are found from the initial condition using the least squares method, which allows their values to be determined with a given accuracy. The solution obtained based on ABC and ASF approximates n → ∞ the exact analytical solution in the form of an infinite series, including trigonometric coordinate functions with coefficients that stabilize exponentially in time. In this case, the eigenvalues determined from the solution of the temporal ordinary differential equation regarding the additional sought function coincide with their exact values at any approximation. The accuracy of the integration constants, determined by the method of least squares, is controlled by the number of approximation points in the range of the auxiliary variable's variation. It should be noted that the additional boundary conditions considered in this work hold true for any other method of obtaining a solution to the problem under consideration, including the exact method, as can be verified by direct substitution. Therefore, their introduction does not distort the original mathematical formulation of the problem but only significantly simplifies the process of obtaining its analytical solution.

Quasi-periodic solutions for differential equations with an elliptic equilibrium under delayed perturbation

We employ the Craig-Wayne-Bourgain (CWB) method to construct quasi-periodic solutions for the nonlinear delayed perturbation equations. The linearized equation at each step of the iterations is a differential-difference equation on the torus. We shall combine the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, which generalizes the applicability of the CWB method. As an application, we study the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay.

Results on the growth of solutions of complex linear differential equations with meromorphic coefficients

The purpose of this paper is the study of the growth of solutions of higher order linear differential equations \(f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left(z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right)f^{\left( k-1\right) }+\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right) }+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=0\) and \(f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left(z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right)f^{\left( k-1\right) }+\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right)}+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=F\left( z\right),\) where \(A_{j,i}\left( z\right) \left( \not\equiv 0\right) \left(j=0,...,k-1;i=1,2\right) ,\) \(F\left( z\right) \) are meromorphic functions of finite order and \(P_{j}\left( z\right) ,Q_{j}\left( z\right) \) \((j=0,1,...,k-1;i=1,2)\) are polynomials with degree \(n\geq 1\). Under some others conditions, we extend the previous results due to Hamani and Belaïdi [1].

Fixed-point theorems of {F}^{star}- ( psi,phi ) integral-type contractive conditions on 1_{E}-complete multiplicative partial cone metric spaces over Banach algebras and applications

In this paper, we introduce some user-friendly versions of integral-type fixed-point results and give some modifications of the classical Banach contraction principle by constructing a special type of contractive restrictions of integral forms for weak contraction mappings defined on 1_{E}-complete multiplicative partial cone metric spaces over Banach algebras and formulate some existence and uniqueness results regarding the fixed-point theorems using some integrative conditions. Moreover, we validate the significance our results and exploit them to find the unique solution of a fractional nonlinear differential equation of Caputo type, which complements some previously well-known generalizations found in the literature.

On solutions to a novel non-evolutionary integrable 1 + 1 PDE

We investigate real solutions of a C-integrable non-evolutionary partial differential equation in the form of a scalar conservation law where the flux density depends both on the density and on its first derivatives with respect to the local variables. By performing a similarity reduction dictated by one of its local symmetry generators, a nonlinear ordinary differential equation arises that is connected to the Painlevé III equation. Exact solutions are secured and described provided a constraint holds among the coefficients of the original equation. In the most general case, we pinpoint the generation of additional singularities by numerical integration. Then, we discuss the evolution of given initial profiles. Finally, we mention aspects concerning rational solutions with a finite number of poles.

Insight into Functional Boiti–Leon–Mana–Pempinelli Equation and Error Control: Approximate Similarity Solutions

The Boiti–Leon–Mana–Pempinelli Equation (BLMPE) is an essential mathematical model describing wave propagation in incompressible fluid dynamics. In the present manuscript, a novel generalization of the BLMPE is introduced, called herein the functional BLMPE (F-BLMPE), which involves different functions, including exponential, logarithmic and monomaniacal functions. In these cases, the F-BLMPE reduces to an explicit form in the dependent variable. In addition to this, it is worth deriving approximate similarity solutions of the F-BLMPE with constant coefficients using the extended unified method (EUM). In this method, nonlinear partial differential equation (NLPDE) solutions are expressed in polynomial and rational forms through an auxiliary function (AF) with adequate auxiliary equations. Exact solutions are estimated using formal solutions substituted into the NLPDEs, and the coefficients of the AF of all powers are set equal to zero. This approach is valid when the NLPDE is integrable. However, this technique is not valid for non-integrable equations, and only approximate solutions can be found. The maximum error can be controlled by an adequate choice of the parameters in the residue terms (RTs). Multiple similarity solutions are derived, and the ME is depicted in various examples within this work. The results found here confirm that the EUM is an efficient method for solving NLPDEs of the F-BLMPE type.

Adaptive step-size control for global approximation of SDEs driven by countably dimensional Wiener process

In this paper, we deal with the global approximation of solutions of stochastic differential equations (SDEs) driven by countably dimensional Wiener process. Under certain regularity conditions imposed on the coefficients, we show lower bounds for exact asymptotic error behaviour. For that reason, we analyse separately two classes of admissible algorithms: based on equidistant, and possibly not equidistant meshes. Our results indicate that in both cases, decrease of any method error requires significant increase of the cost term, which is illustrated by the product of cost and error diverging to infinity. This is, however, not visible in the finite-dimensional case. In addition, we propose an implementable, path-independent Euler algorithm with adaptive step-size control, which is asymptotically optimal among algorithms using specified truncation levels of the underlying Wiener process. Our theoretical findings are supported by numerical simulation in Python language.

Higher-order time-fractional Sasa–Satsuma equation: Various optical soliton solutions in optical fiber

In this paper, the generalized (3+1)-dimensional nonlinear Sasa–Satsuma model with conformable fractional derivative using the new Kudryashov approach is considered to find a class of novel exact solutions in optical fibers. The acquired new solutions are extract by the hyperbolic functions and exponential function which are assorted as dark, bright, singular, mixed dark–bright, dark–bright, bell-shape, and periodic optical soliton solutions. The contour, three-dimension, two-dimension of various forms of the novel optical solutions are sketched to determine the prominence of the time-fractional generalized (3+1)-dimensional nonlinear Sasa–Satsuma model. In addition, to show the magnitude of the conformable fractional derivative the effect of the conformable fractional order derivative on a class of the new optical solutions are depicted via illustrative graphs. Finally, we found that the present technique is an accurate tool to investigate the analytic solutions of the fractional differential equations. The proposed Sasa–Satsuma model can be applied to the transmission of optical fibers’ ultra-fast pulses.

Fractional-order hybrid functions combining simulated annealing algorithm for solving fractional pantograph differential equations

A new and novel numerical method has been developed based on the fractional-order hybrid functions combining simulated annealing (SA) algorithm for the solutions of fractional pantograph delay differential equations. First of all, a fractional-order hybrid of block-pulse functions and Chebyshev polynomials (FOHBPCs) is defined. With the aid of regularized beta function, the exact formulas of FOHBPCs are derived under the definition of Riemann–Liouville fractional integral. And then, by the properties of FOHBPCs and the exact formulas together with the collocation method, the problem under consideration is simplified into algebraic equations, which are solved by Gaussian elimination method and Picard iteration method for linear and nonlinear cases, respectively. The error analysis of the proposed method is investigated. Due to the importance of the parameter α in the fractional-order hybrid functions method, SA algorithm is considered to find the optimal parameter α. Finally, the effectiveness and applicability of the suggested method are verified through some numerical examples.

The effects of thermal radiation and heat source/sink on the flow and heat transfer characteristics of a hybrid nanofluid over a vertical stretching cylinder: Regression analysis

Originality: A novel category of working fluids, consisting of two substantial components diffused in a conventional fluid, has been identified and investigated widely in recent years. These types of fluids are called hybrid nanofluids. Problem statement: A wide range of engineering and industrial structures, including heat-transferring components, energy production, extrusion procedures, engine cooling purposes, thermal structures, thermal exchangers, chemical procedures, manufacturing processes and hybrid power plants, have been proposed for use with nanomaterials with improved thermal properties. These nanomaterial-based applications hold the promise of improved performance and efficiency in a variety of technological and industrial processes. The heat transmission and magnetohydrodynamic stagnation significance flow of hybrid nanofluids Fe3O4–ZrO2 and Fe3O4/water, the form factor of a stretched cylinder under the influence of heat production, nonlinear thermal radiation and nanoparticle volume fractions have been investigated in this study. Methodology: Utilizing proper similarity transformations, the processes of partial differential equations are further transformed into nondimensional solutions of ordinary differential equations. The bvp4c approach is employed to achieve a numerical solution. The flow and temperature profiles are displayed as a function of the contained factors. Graphs show the effects of changing the physical characteristics involved. Tables emphasize the skin friction factor and Nusselt numbers. Results: The temperature profile of fluids diminished due to an increment in the values of the temperature relaxation parameter and Eckert number. When the porosity factor is increased the temperature of fluids is improved. The effects of streamlines for various components are discussed. The 3D surface, contour plots and residual plots for various factors have also been investigated. Applications: Hybrid nanofluids have the potential to improve heat transfer efficiency in a variety of technical applications, including cooling structures, heat exchangers and thermal energy storage systems.

A Systematic Approach to Delay Functions

We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions.