Higher-order Differential Equations Research Articles

Novel Approaches to Existence and Uniquenessin Nonlinear Higher-Order Differential Equations

The study of nonlinear higher-order differential equations presents significant challenges in terms of existence and uniqueness of solutions. This paper explores novel approaches to addressing these challenges, focusing on recent advancements and methodologies that offer new perspectives on these classical problems. We investigate advanced techniques including topological methods, functional analysis, and computational approaches to enhance our understanding of existence and uniqueness in nonlinear higher-order differential equations. By reviewing the latest literature and integrating new findings, this paper aims to provide a comprehensive overview of current research trends and future directions in this area.

Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics

Physics-informed neural network has emerged as a promising approach for solving partial differential equations. However, it is still a challenge for the computation of structural mechanics problems since it involves solving higher-order partial differential equations as the governing equations are fourth-order nonlinear equations. Here we develop a multi-level physics-informed neural network framework where an aggregation model is developed by combining multiple neural networks, with each one involving only first-order or second-order partial differential equations representing different physics information such as geometrical, constitutive, and equilibrium relations of the structure. The proposed framework demonstrates a remarkable advancement over the classical neural networks in terms of the accuracy and computation time. The proposed method holds the potential to become a promising paradigm for structural mechanics computation and facilitate the intelligent computation of digital twin systems.

Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation\begin{align*}\gamma^{iv} (\omega) + \rho_1 \gamma{'''} (\omega)+ \rho_2 \gamma{''} (\omega) + \rho_3 \gamma' (\omega) + \rho_4 \gamma(\omega) = \chi(\omega),\end{align*}where $\gamma \in c^4 [\alpha,\beta], \chi \in [\alpha,\beta]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.

On a Poincaré-Perron problem for high order differential equations

On a Poincaré-Perron problem for high order differential equations

Chebyshev polynomial derivative-based spectral tau approach for solving high-order differential equations

Chebyshev polynomial derivative-based spectral tau approach for solving high-order differential equations

Numerical study of hybrid nanofluid and thermal transport in sun-powered energy ship within the application of parabolic trough solar collectors

PurposeRecent advancements in technology have led to the exploration of solar-based thermal radiation and nanotechnology in the field of fluid dynamics. Solar energy is captured through sunlight absorption, acting as the primary source of heat. Various solar technologies, such as solar water heating and photovoltaic cells, rely on solar energy for heat generation. This study focuses on investigating heat transfer mechanisms by utilizing a hybrid nanofluid within a parabolic trough solar collector (PTSC) to advance research in solar ship technology. The model incorporates multiple effects that are detailed in the formulation.Design/methodology/approachThe mathematical model is transformed using suitable similarity transformations into a system of higher-order nonlinear differential equations. The model was solved by implementing a numerical procedure based on the Wavelets and Chebyshev wavelet method for simulating the outcome.FindingsThe velocity profile is reduced by Deborah's number and velocity slip parameter. The Ag-EG nanoparticles mixture demonstrates less smooth fluid flow compared to the significantly smoother fluid flow of the Ag-Fe3O4/EG hybrid nanofluids (HNFs). Additionally, the Ag-Ethylene Glycol nanofluids (NFs) exhibit higher radiative performance compared to the Ag-Fe3O4/Ethylene Glycol hybrid nanofluids (HNFs).Practical implicationsAdditionally, the Oldroyd-B hybrid nanofluid demonstrates improved thermal conductivity compared to traditional fluids, making it suitable for use in cooling systems and energy applications in the maritime industry.Originality/valueThe originality of the study lies in the exploration of the thermal transport enhancement in sun-powered energy ships through the incorporation of silver-magnetite hybrid nanoparticles within the heat transfer fluid circulating in parabolic trough solar collectors. This particular aspect has not been thoroughly researched previously. The findings have been validated and provide a highly positive comparison with the research papers.

Chaotic systems based on higher-order oscillatory equations

This paper discusses the design process toward new lumped chaotic systems that originates in higher-order ordinary differential equations commonly used as description of ideal oscillators. In investigated third-order case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourth-order oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both “superelements” are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Exploring solutions to the fractional Boiti–Leon–Manna–Pempinelli equation characterizing wave dynamics in incompressible fluids

In this study, the (3 + 1)-dimensional space-time fractional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated utilizing the Kudryashov method (KM) and the modified Kudryashov method (MKM). These two efficient methods are implemented to acquire exact closed-form solutions to the considered fractional BLMP equation, and novel solitary wave solutions are constructed involving hyperbolic functions, and exponential rational functions with arbitrary constant parameters. With the aid of a unique wave transformation, the nonlinear fractional differential equation is transformed into an ordinary differential equation. In this unique wave transformation, the conformable fractional derivative is considered to which the chain rule is applied. The obtained solutions are indeed beneficial for analyzing the dynamic behavior of the fractional BLMP equation in describing fluid propagation. A few three-dimensional representations of the attained exact analytic solutions are offered by accounting for the proper selection of the appropriate parameters with the help of mathematical software. As a result, novel soliton solutions, such as the anti-kink type and singular kink type waves, are acquired from the attained solutions. This approach is relatively straightforward and efficient, making it possible to utilize it to obtain exact solutions for higher-order nonlinear partial differential equations.

Optimization of heat transfer properties in micropolar ternary (Al2O3 + TiO2 + SiO2/water) nanofluid with porosity and magnetic field effects

Decades of study provide substantial evidence supporting that nanofluids possess improved heat transmission capabilities. Optimal qualities of nanoliquids may be achieved by manipulating the volume concentration of the nanoparticles. Nevertheless, this method has constraints due to the balance between negative and positive viscosity. A recent advancement has been produced in creating and thoroughly examining a particular form of hydraulic fluid that incorporates three solid nanoparticles scattered inside an existing fluid. This improvement aims to tackle this limitation. This study investigates the characteristics of a micro-rotational incompressible micropolar ternary hybrid nanofluid in a porous media. It considers the influence of magnetohydrodynamics, Darcy-Forchheimer, and linear thermal radiations on a stretched surface. The nanofluid consists mostly of water, which is blended with titanium oxide, aluminum oxide, and silicon oxide nanoparticles. Using dimensionless similarity transformations, the collection of higher-order non-dimensional partial differential equations is converted into higher-order ordinary differential equations. The BVP4C technique in MATLAB software is used to solve these equations and assess the graphical and tabular outcomes. The graphical analysis examines the influence of various physical characteristics. The heat transfer of a ternary hybrid nanofluid is significantly affected by many factors, including the Forchheimer coefficient, Prandtl number, thermal radiation parameter, and magnetic field parameters, as seen. The research demonstrates that an increase in the magnetic field and Darcy-Forchheimer parameters reduces linear momentum. Conversely, increased porosity, magnetic field, and radiation parameters increase the temperature profile.

Effective Method for Solving Higher-Order Fractional Differential Equations Using Spline Functions

In this paper, appropriate spline functions in polynomial form are outlined and applied to solve higher-order linear fractional differential equations (H-OLFDEs). Several techniques have been proposed to solve this type of equations. A description of the projected methodology is first introduced. The method involves transforming the H-OLFDE of order α, with given initial conditions, into a system of fractional differential equations with , denoting the order of the Caputo fractional derivative for each equation i in the transformed system, where the number of equations in the resulting system is equal to that of initial conditions. The numerical results obtained using this strategy demonstrate substantial agreement with the precise analytical solutions. The results in this paper affirm the robustness and ease of application of the proposed technique.

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Bundle-specific tractogram distribution estimation using higher-order streamline differential equation

Streamline tractography locally traces peak directions extracted from fiber orientation distribution (FOD) functions, lacking global information about the trend of the whole fiber bundle. Therefore, it is prone to producing erroneous tracks while missing true positive connections. In this work, we propose a new bundle-specific tractography (BST) method based on a bundle-specific tractogram distribution (BTD) function, which directly reconstructs the fiber trajectory from the start region to the termination region by incorporating the global information in the fiber bundle mask. A unified framework for any higher-order streamline differential equation is presented to describe the fiber bundles with disjoint streamlines defined based on the diffusion vectorial field. At the global level, the tractography process is simplified as the estimation of BTD coefficients by minimizing the energy optimization model, and is used to characterize the relations between BTD and diffusion tensor vector under the prior guidance by introducing the tractogram bundle information to provide anatomic priors. Experiments are performed on simulated Hough, Sine, Circle data, ISMRM 2015 Tractography Challenge data, FiberCup data, and in vivo data from the Human Connectome Project (HCP) for qualitative and quantitative evaluation. Results demonstrate that our approach reconstructs complex fiber geometry more accurately. BTD reduces the error deviation and accumulation at the local level and shows better results in reconstructing long-range, twisting, and large fanning tracts.

Special function models of indecomposable sl(2) representations: the Laguerre case

In this paper, we point out connections between certain types of indecomposable representations of sl(2) and generalizations of well-known orthogonal polynomials. Those representations take the form of infinite dimensional chains of weight or generalised weight spaces, for which the Cartan generator acts in a diagonal way or via Jordan blocks. The other generators of the Lie algebras sl(2) act as raising and lowering operators but are now allowed to relate the different chains as well. In addition, we construct generating functions, we calculate the action of the Casimir invariant and present relations to systems of non-homogeneous second-order coupled differential equations. We present different properties as higher-order linear differential equations for building blocks taking the form of one variable polynomials. We also present insight into the zeroes and recurrence relations.

Boundary value problems for integro-differential and singular higher-order differential equations

Abstract We investigate third-order strongly nonlinear differential equations of the type ( Φ ( k ( t ) u ″ ( t ) ) ) ′ = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t){u}^{^{\prime\prime} }\left(t)))^{\prime} =f\left(t,u\left(t),u^{\prime} \left(t),{u}^{^{\prime\prime} }\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], where Φ \Phi is a strictly increasing homeomorphism, and the non-negative function k k may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like ( Φ ( k ( t ) v ′ ( t ) ) ) ′ = f t , ∫ 0 t v ( s ) d s , v ( t ) , v ′ ( t ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t)v^{\prime} \left(t)))^{\prime} =f\left(t,\underset{0}{\overset{t}{\int }}v\left(s){\rm{d}}s,v\left(t),v^{\prime} \left(t)\right),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], for which we provide existence results for various types of boundary conditions, including periodic, Sturm-Liouville, and Neumann-type conditions.

Comparison Theorems for Oscillation of Higher-Order Neutral Delay Differential Equations

In this work, we study the oscillatory features of a class of neutral differential equations with multiple delays. We present novel oscillation conditions for this equation by using a comparison method. We create conditions that exclude the positive solution of the examined equation. Using the feature of symmetry between non-oscillatory solutions (positive and negative), these conditions also exclude negative solutions without adding additional restrictions. In our study, we take into account the even and odd cases of the order of the equation. Furthermore, we study the asymptotic properties in three different cases of functional coefficients. Our results are a fundamental extension and generalization of previous relevant findings, and this fact has been demonstrated through comparisons.

Lyapunov-type inequalities for higher-order Caputo fractional differential equations with general two-point boundary conditions

Lyapunov-type inequalities for higher-order Caputo fractional differential equations with general two-point boundary conditions

A Well-Conditioned Spectral Integration Method for High-Order Differential Equations with Variable Coefficients

A Well-Conditioned Spectral Integration Method for High-Order Differential Equations with Variable Coefficients

Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients

Abstract In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients.

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.