Third-order Differential Equation Research Articles

Numerical approximation of the 3rd order pseudo-parabolic equation using collocation technique

This study presents a novel numerical approach for approximating the solution of third-order pseudo-parabolic partial differential equations (PDEs), which exhibit both parabolic and hyperbolic characteristics. The proposed method employs a cubic trigonometric tension B-spline collocation technique for spatial discretization, offering greater flexibility and accuracy compared to traditional spline methods. For time discretization, the finite difference method (FDM) is used, ensuring computational efficiency. Unlike many existing methods, our approach is tailored to handle the complexity of third-order equations while maintaining stability and accuracy over large-scale problems. The method’s unconditional stability is confirmed through a detailed von Neumann stability analysis, making it particularly robust for long-term simulations. Two illustrative examples are presented to demonstrate the method’s superior accuracy and flexibility in handling complex boundary conditions, as well as its ability to manage large-scale problems without requiring restrictive time steps. Compared to the existing methods, the combination of trigonometric tension B-splines with FDM proves to be a powerful and reliable tool for solving higher-order pseudo-parabolic equations.

New oscillatory criteria for third-order differential equations with mixed argument

Abstract In this paper, we offer new technique for investigation of the third-order linear differential equation with mixed argument y ‴ ( t ) + p ( t ) y ( τ ( t ) ) = 0. $$\begin{array}{} \displaystyle y'''(t)+p(t)y(\tau(t))=0. \end{array}$$ (E) We establish new criteria for property A of (E) which fulfill the gap in the oscillation theory.

Investigation of the Oscillatory Behavior of the Solutions of a Class of Third-Order Delay Differential Equations with Several Terms

In this paper, we address the study of the oscillatory properties of the solutions of a class of third-order delay differential equations. The primary objective of this study is to provide new relationships that can be employed to obtain criteria for excluding increasing positive solutions and decreasing positive solutions so that the resulting criteria are easier to apply than other criteria that have appeared in the literature. We have obtained new oscillation criteria that hold up more robustly upon application. Some examples are presented to illustrate the significance of our main findings.

Self-Similar Solutions of a Multidimensional Degenerate Partial Differential Equation of the Third Order

When studying the boundary value problems’ solvability for some partial differential equations encountered in applied mathematics, we frequently need to create systems of partial differential equations and explicitly construct linearly independent solutions explicitly for these systems. Hypergeometric functions frequently serve as solutions that satisfy these systems. In this study, we develop self-similar solutions for a third-order multidimensional degenerate partial differential equation. These solutions are represented using a generalized confluent Kampé de Fériet hypergeometric function of the third order.

A Modified Bubnov-Galerkin Method for Solving Boundary Value Problems with Linear Ordinary Differential Equations

Introduction. The paper considers the solution of boundary value problems on an interval for linear ordinary differential equations, in which the coefficients and the right-hand side are continuous functions. The conditions for the orthogonality of the residual equation to the coordinate functions are supplemented by a system of linearly independent boundary conditions. The number of coordinate functions m must exceed the order n of the differential equation. Materials and Methods. To numerically solve the boundary value problem, a system of linearly independent coordinate functions is proposed on a symmetric interval [−1,1], where each function has a unit Chebyshev’s norm. A modified Petrov-Galerkin method is applied, incorporating linearly independent boundary conditions from the original problem into the system of linear algebraic equations. An integral quadrature formula with twelfth-order error is used to compute the scalar product of two functions. Results. A criterion for the existence and uniqueness of a solution to the boundary value problem is obtained, provided that n linearly independent solutions of the homogeneous differential equation are known. Formulas are derived for the matrix coefficients and the coefficients of the right-hand side in the system of linear algebraic equations for the vector expansion of the solution in terms of the coordinate function system. These formulas are obtained for second- and third-order linear differential equations. The modified Bubnov-Galerkin method is formulated for differential equations of arbitrary order. Discussion and Conclusions. The derived formulas for the generalized Bubnov-Galerkin method may be useful for solving boundary value problems involving linear ordinary differential equations. Three boundary value problems with second- and third-order differential equations are numerically solved, with the uniform norm of the residual not exceeding 10–11.

Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

Absolutely stable difference scheme for the delay partial differential equation with involution and Robin boundary condition

This paper examines the initial value problem for a third-order delay partial differential equation with involution and Robin boundary condition. We construct a first-order accurate difference scheme to obtain the numerical solution for this equation. Illustrative numerical results are provided.

On the existence of a positive solution to a boundary value problem for a third-order nonlinear functional differential equation with an integral boundary condition at one of the ends of the segment

The article studies the existence of positive solutions on the segment [0,1] of a two-point boundary value problem for one nonlinear third-order functional differential equation with an integral boundary condition at one of the ends of the segment. Using the Go–Krasnoselsky fixed point theorem and some properties of the Green's function of the corresponding differential operator, sufficient conditions for the existence of at least one positive solution to the problem under consideration are obtained. An example is given to illustrate the results obtained.

Existence and bounds for Kneser-type solutions to noncanonical third-order neutral differential equations

This article focuses on the existence and asymptotic behavior of Kneser-type solutions to third-order noncanonical differential equations with a delay or advanced argument in the neutral term $$ \Big(r_2(t)\big(r_1(t)z'(t)\big)'\Big)'+g(t)x(t)=0, $$ where \( z(t)=x(t)+p(t)x(\tau(t))\). This equation is transformed into a canonical equation, which reduces the number of classes of positive solutions from 4 to 2. This is done without assuming extra conditions, and greatly simplifies the process of obtaining conditions for the existence of Kneser-type solutions. Also we obtain lower and upper bounds for these solutions, and obtain their rate of convergence to zero. Two examples are provided to illustrate our main results, one with a delay neutral term, and one with an advanced neutral term. For more information see https://ejde.math.txstate.edu/Volumes/2024/55/abstr.html

A qualitative investigation of a system of third-order difference equations with multiplicative reciprocal terms

In this paper, we study the system of third-order difference equations \begin{equation*} x_{n+1}=a+\frac{a_{1}}{y_{n}}+\frac{a_{2}}{y_{n-1}}+\frac{a_{3}}{y_{n-2}}% ,\quad y_{n+1}=b+\frac{b_{1}}{x_{n}}+\frac{b_{2}}{x_{n-1}}+\frac{b_{3}}{% x_{n-2}},\quad n\in \mathbb{N}_{0}, \end{equation*}% where the parameters $a$, $a_{i}$, $b$, $b_{i}$, $i=1,2,3$, and the initial values $x_{-j}$, $y_{-j}$, $j=0,1,2$, are positive real numbers. We first prove a general convergence theorem. By applying this convergence theorem to the system, we show that positive equilibrium is a global attractor. We also study the local asymptotic stability of the equilibrium and show that it is globally asymptotically stable. Finally, we study the invariant set of solutions.

On a class of third-order dissipative differential operator with distributional potentials

This paper aims to investigate a class of boundary value problems generated by a third-order differential equation with distributional potentials and some dissipative boundary conditions. We first construct an operator related to the problem, then prove that the operator is dissipative and show some eigenvalue properties of this operator. Further using Krein’s theorem we prove the completeness theorems of the operator.

Statistical mechanics of passive Brownian particles in a fluctuating harmonic trap.

We consider passive Brownian particles trapped in an "imperfect" harmonic trap. The trap is imperfect because it is randomly turned off and on, and as a result particles fail to equilibrate. Another way to think about this is to say that a harmonic trap is time dependent on account of its strength evolving stochastically in time. Particles in such a system are passive and activity arises through external control of a trapping potential, thus, no internal energy is used to power particle motion. A stationary Fokker-Planck equationof this system can be represented as a third-order differential equation, and its solution, a stationary distribution, can be represented as a superposition of Gaussian distributions for different strengths of a harmonic trap. This permits us to interpret a stationary system as a system in equilibrium with quenched disorder.

Solvability of functional third-order problems of Ambrosetti–Prodi-type

This work presents an Ambrosetti–Prodi alternative for functional problems composed of a fully third-order differential equation with two types of functional boundary conditions. The discussion of existence and non-existence of solution is obtained in a more general case, and the multiplicity of solution is done with restrictive boundary conditions-The main arguments are based on the lower and upper solutions method, together with the Leray–Schauder topological degree theory. We stress that the multiplicity situation requires different speed growths on the variables.An example illustrates the results’ applicability and shows a technique to estimate the bifurcation values of the parameter.

Unveiling advection–dominated interactions: Efficacy of neural networks in natural systems modelling

Studying the interactions between advection and dispersion in natural systems, especially in cases where advection predominates, are important because it is necessary to accurately model physical phenomena in domains like hydrology, atmospheric science, environmental engineering, etc. Conventional analytical and numerical methods often have drawbacks, such as high computational costs and challenges in accurately modeling complex behaviors. Improved simulation and a deeper understanding of these interactions are made possible by neural networks’ capacity to learn from big datasets and simulate complex relationships. To demonstrate that deep neural networks effectively capture scenarios within physical processes where advection plays a dominant role, the effectiveness of applying deep neural networks to a third-order dispersive partial differential equation incorporating advection and dispersion terms has been investigated in this study.

Boundary value problem with a spectral parameter

In this paper we consider one class of irregular boundary value problems for ordinary differential equations of third order. Their defective coercivity is investigated and the defect of coercivity is found and the area of defective coercivity is determined, which proves the existence of a unique solution of the problem. An estimation of the solution has obtained.

Stochastic neuro-swarming intelligence paradigm for the analysis of magneto-hydrodynamic Prandtl–Eyring fluid flow with diffusive magnetic layers effect over an elongated surface

In recent years, the integration of stochastic techniques, especially those based on artificial neural networks, has emerged as a pivotal advancement in the field of computational fluid dynamics. These techniques offer a powerful framework for the analysis of complex fluid flow phenomena and address the uncertainties inherent in fluid dynamics systems. Following this trend, the current investigation portrays the design and construction of an important technique named swarming optimized neuro-heuristic intelligence with the competency of artificial neural networks to analyze nonlinear viscoelastic magneto-hydrodynamic Prandtl–Eyring fluid flow model, with diffusive magnetic layers effect along an extended sheet. The currently designed computational technique is established using inverse multiquadric radial basis activation function through the hybridization of a well-known global searching technique of particle swarm optimization and sequential quadratic programming, a technique capable of rapid convergence locally. The most appropriate scaling group involved transformations that are implemented on governing equations of the suggested fluidic model to convert it from a system of nonlinear partial differential equations into a dimensionless form of a third-order nonlinear ordinary differential equation. The transformed/reduced fluid flow model is solved for sundry variations of physical quantities using the designed scheme and outcomes are matched consistently with Adam's numerical technique with negligible magnitude of absolute errors and mean square errors. Moreover, it is revealed that the velocity of the fluid depreciates in the presence of a strong magnetic field effect. The efficacy of the designed solver is depicted evidently through rigorous statistical observations via exhaustive numerical experimentation of the fluidic problem.

Existence, Uniqueness, and Stability of a Nonlinear Tripled Fractional Order Differential System

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs).

Advanced color image encryption using third-order differential equations and three-dimensional logistic map

Image encryption stands out as a crucial technique employed to securely transmit images across the Internet. In this paper, we introduce a novel algorithm for encrypting color images. The algorithm is built upon the principles of differential equations, XOR operations, and chaotic maps. First, the plain image is three-dimensional pixel shuffled via a logistic map. Afterward, the differential equations are used as a mathematical tool for encrypting images. The third-order ordinary differential equations are used to encrypt the shuffled images. The color values of the plain image are considered coefficients for the independent variable X. Subsequently, an alternate matrix of the same size is generated using a three-dimensional logistic map, taking into account its color values as the exponents linked to the independent variable X. A set of third-order differential equations emerged, containing an equivalent number of elements as the color values present in the plain image. This set of differential equations is addressed in the following manner: combining XOR and integration three times with respect to the independent variable X for each set of obtained differential equations while treating the integration constant as zero. Ultimately, a set of ordinary equations involving the independent variable X is derived, where the coefficients of X represent color values for the cipher image. The results from experiments and the security analysis affirm the resilience of the proposed algorithm against established security attacks. It exhibits a substantial key space, heightened key sensitivity, and a strong encryption effect.

Study the Effect of Suction-Injection parameters Numerically and Analytically for Magneto hydrodynamic Jeffery Hamel Fluid Flow Problem

This paper processes numerically and analytically the magnetohydrodynamic flow among two porous solid plate intersections at an angle or through non-parallel porous walls, which can be interpreted as a mix of Jeffery Hamel fluid flow additives from suction-injection parameters. In fluid mechanics, the equations of governing, which are transferred to the non-linear ordinary differential equation of third order, are modeled instead of the conventional Navier-Stokes equations by Maxwell's electromagnetism. Results obtained from the DTM and the numerical method BVP4c are compared to those resulting from the optimal methods (ODTM), which consist of the collocation differential transform method (CDTM), the sub-domain differential transform method (SDTM), the least squares differential transform method (LSDTM), and the Galerkin differential transform method (GDTM). A comparison is made between the approximate analytical and numerical solutions for different parameters like open angles (ε), Reynolds numbers (Re), injection parameters (S), and Hartmann numbers (Ha). The results demonstrate that the two approaches are very similar.

On the existence and uniqueness of a positive solution to a boundary value problem for one nonlinear functional differential equation of third order

On the existence and uniqueness of a positive solution to a boundary value problem for one nonlinear functional differential equation of third order