Solutions Of Differential Equations Research Articles

The improved residual power series method for the solutions of higher-order linear and nonlinear boundary value problems

AbstractIRPSM is the extended form of RPSM that gives an approximate solution to boundary value problems without requiring an exact solution. This method creates a truncated series for the determination of the missing initial conditions. The present study is conducted to evaluate semi-numerical solutions to differential equations using the Improved Residual Power Series Method (IRPSM). Our main objective is to check whether the proposed scheme is efficient for solving such equations or not. The solution of differential equations has been approximated using truncated residual power series. The method is used to solve differential equations for higher-order linear and non-linear boundary value problems. Absolute errors for the solved problems are calculated to ensure accuracy. It is also compared with some known methods, like the Differential Transform Method (DTM) and the Homotopy Perturbation Method (HPM). For the calculations, Mathematica 12.0 software is used. The computed results are compared to the exact solutions as well as the available results in the literature. The results showed that the results obtained by IRPSM are closely related to the exact solution when compared to the DTM and HPM results.

Unsteady MHD dusty fluid flow over a cone in the vicinity of porous medium: a numerical study

This research presents an unsteady free convection magnetohydrodynamics (MHD) dusty fluid flow over a vertical cone enclosed inside a porous medium. The Crank–Nicolson approach is used to get the numerical solutions for these non-linear partial differential equations (PDEs). The velocity and temperature distributions are graphed numerically for a wide range of physical parameters to highlight important characteristics of the solutions. The results showed that increasing the porosity parameter leads to a rise in the velocity profile for the fluid and dust phases, while doing the reverse with the rise of the magnetic parameter, fluid-particle interaction parameter, and mass concentration of the particle phase parameter. Additionally, it is shown that the temperature profile rises as the magnetic parameter and the mass concentration of particle phase parameter grow, whereas the fluid-particle interaction parameter and porosity parameter show the opposite tendency.

Bending of Lloyd’s mirror to eliminate the period chirp in the fabrication of diffraction gratings

We present a new technique to prevent the detrimental period chirp that appears in optical gratings fabricated by laser interference lithography (LIL). The idea is to bend the Lloyd’s mirror in the lithographic setup to eliminate the period chirp already at the step of the grating’s exposure. A new mathematical model was developed to describe the required bending geometry of the mirror. It is shown that this geometry can be described by multiple cross-sections of the mirror, each obtained by the solution of an implicit first-order differential equation. The proposed approach is illustrated on the basis of a concrete example. By slightly bending the Lloyd’s mirror (by ≈ 3.5 mm of maximum deflection over an area of 142 mm × 215 mm) the period chirp of the exposed grating can be eliminated completely.

Exact solutions of the Landau–Ginzburg–Higgs equation utilizing the Jacobi elliptic functions

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations.

A new version of trial equation method for a complex nonlinear system arising in optical fibers

In this study, the dissipation problem of nonlinear pulse in mono mode optical fibers which is governed by the Fokas system (FS) is considered. The solutions of this system have an important role in comprehending the different wave structures in physical settings. Therefore, a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the FS. The advantage of the NVTEM is to use different root possibilities of a polynomial which shape the solutions of the related model. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various exact wave solutions to the FS are obtained such as rational function, exponential function, hyperbolic function, and Jacobi elliptic function solutions. Thus, we have revealed solutions featly which are unlike the wave solutions previously found by other analytical methods. The present results depict the formation and development of such waves and their interactions. The exhibition of the solutions is given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant in achieving different wave solutions of many nonlinear differential equations in the field of optics.

Topology and Dynamics of Transcriptome (Dys)Regulation

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let us delve into the mathematical intricacies of these sequences. We have previously devised a mathematical approach for defining a “healthy” sequence. This sequence is characterized by having at most four distinct nucleotides (denoted as nt≤4). It serves as the generator of a group denoted as fp. The desired properties of this sequence are as follows: fp should be close to a free group of rank nt−1, it must be aperiodic, and fp should not have isolated singularities within its SL2(C) character variety (specifically within the corresponding Groebner basis). Now, let us explore the concept of singularities. There are cubic surfaces associated with the character variety of a four-punctured sphere denoted as S24. When we encounter these singularities, we find ourselves dealing with some algebraic solutions of a dynamical second-order differential (and transcendental) equation known as the Painlevé VI Equation. In certain cases, S24 degenerates, in the sense that two punctures collapse, resulting in a “wild” dynamics governed by the Painlevé equations of an index lower than VI. In our paper, we provide examples of these fascinating mathematical structures within the context of miRNAs. Specifically, we find a clear relationship between decorated character varieties of Painlevé equations and the character variety calculated from the seed of oncomirs. These findings should find many applications including cancer research and the investigation of neurodegenative diseases.

Weak solutions to coupled quadratic forward backward stochastic differential equations and Sobolev solutions to their related partial differential equations

We establish existence and uniqueness of a weak (in law sense) solution to coupled forward–backward stochastic differential equations (FBSDEs), with possibly discontinuous diffusion coefficient. We assume that the coefficient of the backward component is of quadratic growth. We first prove the existence of a Sobolev solution for the related partial differential equation (PDE) in the Sobolev space , by using compactness arguments. Next, we use Itô–Krylov's formula to get the existence of weak solution to our considered FBSDE. For the uniqueness part, we first establish the weak uniqueness of the FBSDEs then deduce the uniqueness of its related PDE by using the Feynmann–Kac formula.

The solution of a complex integro differential equation in the right half-plane via an infinite plate weakened by a curvilinear hole

The solution of a complex integro differential equation in the right half-plane via an infinite plate weakened by a curvilinear hole

New analytical technique for prototype closed form solutions of certain nonlinear partial differential equations

This work proposes a novel modified (G’/G2)-expansion method to construct the exact travelling wave solutions of three famous nonlinear partial differential equations namely, nanoionic currents along microtubules equation, Benney-Luke equation and the generalized Hirota-Satsuma coupled KdV system. In the result, these solutions can be divided into different types i.e. trigonometric, rational and exponential solutions. The computational tool Maple is used to arrive at these solutions. Applying the novel modified (G’/G2)-expansion approach yields results that are trustworthy, easy to understand, and useful in a variety of disciplines including physics, biology, finance, engineering, statistics, and more. Software is used to create the two- and three-dimensional graphs. The comparison of the obtained solutions is also given in the form of Tables 1, 2 and 3, to show the novelty of our obtained solutions.

Review of Adams-Bashforth method for numerical solution of first order ordinary differential equations

This project work presents a comprehensive review of the Adams-Bashforth method for the numerical solution of explicit first-order ordinary differential equations (ODEs). The study begins with a historical overview of the development of ordinary differential equations (ODEs), tracing back to the seminal works of Newton, Leibniz, and their contemporaries. The evolution of differential equations as a distinct mathematical discipline and their wide-ranging applications across various fields are discussed. Furthermore, the paper provides an in-depth analysis of the Adams-Bashforth method, a prominent numerical technique for solving ODEs. The method is derived and analyzed using Mathematica Software, demonstrating its efficacy and accuracy in approximating solutions to ODEs. A comparison with existing methods such as the Euler method and the Runge-Kutta method highlights the advantages of the Adams-Bashforth method, particularly in terms of computational efficiency and accuracy. Moreover, fundamental concepts related to ODEs, including the initial value problem, existence, and uniqueness of solutions, are explored in the context of numerical solution methods. The properties of linear multistep methods and their relevance to solving ODEs efficiently are also discussed. In conclusion, this project work provides valuable insights into the Adams-Bashforth method as a powerful tool for the numerical solution of first-order ODEs. By leveraging modern computational tools such as Mathematica Software, the method offers a practical and efficient approach to solving complex differential equations encountered in various scientific and engineering applications.

Existence of a periodic solution for a state-dependent delay differential equation with a singular perturbation

The idea of this article came from the works dones by Khung and Smith on the one hand in [2] for  and Arino and Al in [1] for  on the other hand, to proves the existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard. I proposed the following question: Do the result remain valid if  1 ? by the same method with using the homotopy well defined link between the application of Khung and Smith and the application of Arino and  and indeed, I came to prove existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard, therefore the result obtained in this work is generalization of two special cases. A State-dependent Delay Differential equation play important role in recent years to modeling of several phenomenon, in dynamics of populations, in biology and even in the social life for example: When we take our driving test, we learn that the reaction time of our nervous system when driving is of the order of a few seconds, and that we must take care to maintain a sufficient distance between two cars following each other, from this example, we notice that the delay can be useful. Although this work has added a new addition to the scientific field which is a generalization and linking between two special cases, it remains somewhat limited because it takes special case of Arino and Al. Therefore in the futur, we are thinking about a new work that takes into account the general case of Arino and .

Practical stability of the analytical and numerical solutions of stochastic delay differential equations driven by G-Brownian motion via some novel techniques

In this paper, we focus on stochastic delay differential equations in the G-framework (G-SDDEs). We introduce the practical stability to examine whether the performance of G-SDDE near an unstable equilibrium point is acceptable. We establish a new generalized Gronwall inequality based on which we prove the practical mean-square (PMS) exponential stability of G-SDDE. We also establish the stability equivalence between the discrete and the continuous EM approximations for G-SDDE and then show that the continuous EM approximation can preserve the PMS exponential stability of G-SDDE. One numerical experiment is conducted to confirm our theoretical results.

Existence conditions for 2-periodic solutions to a non-homogeneous differential equations with piecewise constant argument

This paper provides a method of finding 2-periodical solutions for the first-order non-homogeneous differential equations with piecewise constant arguments. All existence conditions are described for 2-periodical solutions and obtained explicit formula for these solutions. An example for the problem that has infinitely many solutions is constructed.

Existence and multiplicity of solutions for fractional differential equations with p-Laplacian at resonance

In this article, we investigate the existence and multiplicity of solutions for a fractional differential equations with p-Laplacian equation at resonance in the \(\psi\)-fractional space \(H^{\alpha, \beta; \psi}_p\). In addition, we show that the energy functional satisfies the Palais-Smale condition. For more information see https://ejde.math.txstate.edu/Volumes/2024/34/abstr.html

Liouville-Green formulas for the fourth order matrix differential operators

In this paper, Liouville-Green analytical approximations for the solutions of the fourth order matrix differential equation are constructed. Using the Gronwall’s inequality, we have also determined the error bounds for the Liouville-Green approximations.

All meromorphic solutions of the autonomous Schwarzian differential equations

AbstractThis paper considers a specific autonomous Schwarzian differential equation given by and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case.

On infinite spectra of oscillation exponents of third order linear differential equations

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.

Runge-Kutta Method Based on the Linear Combination of Arithmetic Mean, Geometric Mean, and Centroidal Mean: A Numerical Solution for the Hybrid Fuzzy Differential Equation

Objectives: To find the absolute error of the Hybrid fuzzy differential equation with triangular fuzzy number since its membership function has a triangular form. Methods: The third order runge-kuttta method based on the linear combination of Arithmetic mean, Geometric mean and Centroidal mean is introduced to solve Hybrid fuzzy differential equation. Seikkala’s derivatives are considered, and Numerical example is given to demonstrate the effectiveness of the proposed method. Findings: The comparative study have been done between the proposed method and the existing third order runge-kutta method based on Arithmetic mean. The proposed method gives better error tolerance than the third order runge-kutta method based on Arithmetic mean. Novelty: In this study a new formula has been developed by combining three means Arithmetic Mean, Geometric Mean and Centroidal Mean using Khattri’s formula and it is solved by the third order runge-kutta method for the first order hybrid fuzzy differential equation. The error tolerance has been calculated for the proposed method and the Arithmetic mean and it is seen that the error tolerance is better for the proposed method. Keywords: Hybrid fuzzy differential equations, Triangular fuzzy number, Seikkala’s derivative, Third order runge-kutta method, Initial value problem

Modeling Radioactive Decay Series Phenomenon Involving First-Order Differential Equation and Visualizing Solutions Using Open-Source Software XCOS

Numerous physical and chemical processes can be accurately described by first-order differential equations and can be solved through analytical methods. The solutions to these equations necessitate the specification of initial or boundary conditions. In the realm of physics and chemistry education, the ability to visualize the solutions of differential equations and comprehend their implications for physical/chemical changes holds significant importance. XCOS, an open-source software tool, serves as a valuable resource for both solving differential equations and graphically representing their solutions. Furthermore, XCOS aids in tackling nonlinear differential equations for which analytical solutions are often elusive. This paper delves into radioactive decay and decay series reactions by first-order differential equations. The paper explores the successful resolution and visualization of the phenomenon involving differential equations through XCOS. This approach extends to the manipulation of various parameters within the equations, further fostering inquiry-based learning.

On Periodic Solutions of a Second-Order Ordinary Differential Equation

On Periodic Solutions of a Second-Order Ordinary Differential Equation