Third-order Differential Equation Research Articles

Image Encryption Algorithm Using Differential Equations

After several prior works in image encryption, where we utilized algebraic mathematical tools within matrix algebra to perform operations that scatter the color values of the image, transforming the image into an encrypted matrix that no longer represents the original image, we decided to explore the field of differential equations as a mathematical tool for image encryption. In this algorithm, we employed third-order ordinary differential equations along with first-degree equations to encode the image. We treated the color values of the original image as coefficients for the independent variable ????. Then, we selected another image as an encryption key, considering its color values as the powers associated with the independent variable ????. This process yielded a system of third-order differential equations, the number of which elements equaled the number of color values in the original image. We approached this system of differential equations by integrating with respect to the independent variable ???? three times for each system of differential equations obtained, treating the integration constant as zero. Ultimately, we obtained a system of ordinary equations with the independent variable ????, where the coefficients of this variable represented alternative values for the color values of the original image. These coefficients represented a scattering of the color values of the original image, and the new matrix formed by these values represented the encryption matrix for the original image. For the purpose of decryption and recovering the original image, we referred to the key image used during the encryption process. We followed these steps: We took the numerical values of the encrypted matrix and converted them into coefficients for the independent variable ????. We considered the color values of the key matrix as the powers associated with the independent variable ???? after adding 3 to them. This step yielded a system of ordinary equations with the number of elements corresponding to the number of color values in the original image. Subsequently, we differentiated each equation in the system of ordinary equations mentioned above. We repeated the differentiation process three times, ultimately obtaining a system of differential equations with the independent variable ????, where the coefficients of this variable represented the color values of the image after decryption. These values matched the color values of the original image prior to decryption, confirming that the error value equaled zero. The results we obtained were excellent and matched high standards of accuracy, which we will discuss later.

Noncanonical Third-Order Advanced Differential Equations of Unstable Type: Oscillation and Property B via Canonical Transform

In this paper, without assuming any extra conditions, the third-order unstable noncaonical advanced differential equation is changed into a canonical form that reduces the number of classes of nonoscillatory solutions into two instead of four. Using comparison and integral averaging method, new sufficient conditions are obtained so that all solutions to have Property B or oscillate. Specific numerical examples are provided to demonstrate the importance and the significance of the main results.

Image Features Hiding by Permutation of Pixels Locations

After completing a scientific research project on image encryption using third-order differential equations, we will focus on designing a new algorithm to conceal the features of the image by exchanging pixel positions in a calculated, precise, and reversible mathematical manner. The exchange will occur in five stages, where the first four stages involve flipping the pixels in a way that resembles paper flipping, causing the pixels to interleave, making their features almost 90% disappear. It is quite natural to significantly increase the disappearance ratio of the image features, reaching up to 99%, by increasing the stages of pixel flipping in different and interleaved positions. However, this is not necessary as we will make the fifth stage of pixel position swapping using a chaotic map, which has the great ability to completely obscure the image features to a ratio of 100%. We applied this algorithm to several images, and the results were excellent. It is worth mentioning that, despite completely hiding the image features, this algorithm is not considered an encryption algorithm because a crucial aspect of encryption, namely the encryption key, is not present. However, it serves as a prelude to the encryption process, we have accomplished through our previous research, where the encryption here is more secure from a security perspective. Additionally, we note that it is possible to enhance this algorithm by adding a key, making it a key-based encryption algorithm, thus obtaining a new encryption algorithm

Colored stochastic multiplicative processes with additive noise unveil a third-order partial differential equation, defying conventional Fokker-Planck equation and Fick-law paradigms

Colored stochastic multiplicative processes with additive noise unveil a third-order partial differential equation, defying conventional Fokker-Planck equation and Fick-law paradigms

Comparing Solutions to the Third-Order Dispersive Partial Differential Equation

In previous decades, many of the practical problems arising in scientific fields such as mathematics, physics, chemistry, biology,and engineering have been related to nonlinear fractional partial differential equations. One of these nonlinear partial differential equations, the third-order dispersive partial differential equation, has been found to have a plethora of useful applications in different fields such as Newtonian fluid mechanics, optimal control, convection diffusion processes, hydrodynamics, and aerodynamics. A special class of solutions has been studied for the third-order dispersive partial differential equation including exact solutions and approximate solutions. The aim of this article is to compare were the Adomian decomposition method, the lines method, an exponential quartic spline and finite difference discretization method, and the non-polynomial spline methodwith the solution of the third-order dispersive partial differential equation. We will conduct a comparison of the stability of the two methods using the Von Neumann stability analysis. In addition, a numerical example will be presented to illustrate the accuracy of these methods.

Solving third order ordinary differential equation with constant coefficients by Adomian decomposition method

The aim of this work is to present an efficient modification of the Adomian Decomposition Method (ADM) for solving third-order ordinary differential equations with constant coefficients. The proposed approach is applicable to both linear and nonlinear problems. To demonstrate the effectiveness of the method, several examples are provided, showcasing its capability to handle both linear and nonlinear ordinary differential equations.

Analysis of solution methods for high order ordinary differential equations used in electrical circuits

This article presents a detailed methodology for applying the most common methods in solving high-order ordinary differential equations in the simulation of electrical circuits. The methods analyzed include the Laplace transform, the direct method in the time domain, the z-transform, the finite difference method, and methods based on difference equations. A detailed development of each of these methods is provided, along with practical examples that demonstrate their application in specific electrical circuits. The examples include: a circuit modeled with a second-order ordinary differential equation, a circuit modeled with a third-order ordinary differential equation, and an electrical network whose modeling results in an eighth-order ordinary differential equation. The article compares the results obtained with each method, using the Laplace transform solution as a reference. A deep analysis of the deviations between the methods is conducted, considering different time steps and parameters, allowing conclusions to be drawn about the effectiveness and accuracy of each approach.

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.

Oscillation of Third-Order Thomas–Fermi-Type Nonlinear Differential Equations with an Advanced Argument

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of Thomas–Fermi-type third-order nonlinear differential equations with advanced argument of the form (a2(t)(a1(t)y′(t))′)′−q(t)yα(σ(t))=0, under the assumptions that ∫t0∞1a2(t)dt<∞ and ∫t0∞1a1(t)dt=∞. The results are achieved by transforming the equation into a canonical-type equation and then applying integral averaging techniques and the comparison method to obtain oscillation criteria for the transformed equation. This in turn will imply the oscillation of the original equation. Several examples are provided to illustrate the significance of the main results.

Solvability of infinite system of third-order differential equations in the sequence space c using Meir–Keeler condensing operator

In this paper, we present an existence result for ω-periodic solutions for an infinite system of third-order differential equations in a classical Banach sequence space c. This result is obtained using techniques related to measures of noncompactness together with the concept of Meir–Keeler condensing operators, considering the respective Green’s function corresponding to the system. To illustrate the significance of our results, we provide some concrete examples.

Novel picosecond wave solutions and soliton control for a higher-order nonlinear Schrödinger equation with variable coefficient

We try to study the propagation of ultrashort optical pulses in dispersion-decreasing fibers using the variable coefficients higher-order nonlinear Schrödinger (VCHNLS) equation. By applying the Jacobi expansion technique combined with a similarity transformation, we reduce the VCHNLS equation to a third-order nonlinear ordinary differential equation. The similarity transformation involves two variables and an exponential term related to the real gain coefficient. Integrability conditions are necessary for this reduction. Using the Jacobi elliptic expansion technique, we find various wave function solutions, including novel periodic waves and previously reported dark solitons. The new Jacobi SN wave solution and its limiting hyperbolic wave function are influenced by the real gain coefficient and the third-order dispersion variable, which control the wave’s shape, amplitude, and phase. This solution is new and gives periodic, kink, bright solitons according to different choices of the controlled variables. Moreover, we also analyze the stability of these solutions.

Energy function of 2D and 3D dynamical systems

It is far well-known that energy function of a two-dimensional autonomous dynamical system can be simply obtained by multiplying its corresponding second-order ordinary differential equation, i.e., its equation of motion by the first time derivative of its state variable. In the nineties, one of us (J.C.S.) stated that a three-dimensional autonomous dynamical system can be also transformed into a third-order ordinary differential equation of motion todays known as jerk equation. Although a method has been developed during these last decades to provide the energy function of such three-dimensional autonomous dynamical systems, the question arose to determine by which type of term, i.e., by the first or second time derivative of their state variable, the corresponding jerk equation of these systems should be multiplied to deduce their energy function. We prove in this work that the jerk equation of such systems must be multiplied by the second time derivative of the state variable and not by the first like in dimension two. We then provide an interpretation of the new term appearing in the energy function and called jerk energy. We thus established that it is possible to obtain the energy function of a three-dimensional dynamical system directly from its corresponding jerk equation. Two and three-dimensional Van der Pol models are then used to exemplify these main results. Applications to Lorenz and Chua’s models confirms their validity.

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.