Solutions Of Differential Equations Research Articles

Multivariate Mittag-Leffler Solution for a Forced Fractional-Order Harmonic Oscillator

The harmonic oscillator is a fundamental physical–mathematical system that allows for the description of a variety of models in many fields of physics. Utilizing fractional derivatives instead of traditional derivatives enables the modeling of a more diverse array of behaviors. Furthermore, if the effect of the fractional derivative is applied to each of the terms of the differential equation, this will involve greater complexity in the description of the analytical solutions of the fractional differential equation. In this work, by using the Laplace method, the solutions to the multiple-term forced fractional harmonic oscillator are presented, described through multivariate Mittag-Leffler functions. Additionally, the cases of damped and undamped free fractional harmonic oscillators are addressed. Finally, through simulations, the effect of the fractional non-integer derivative is demonstrated, and the consistency of the result is verified when recovering the integer case.

Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces

AbstractWe provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.

Analysis of nonlinear fractional-order Fisher equation using two reliable techniques

Abstract In this article, the solution to the time-fractional Fisher equation is determined using two well-known analytical techniques. The suggested approaches are the new iterative method and the optimal auxiliary function method, with the fractional derivative handled in the Caputo sense. The obtained results demonstrate that the suggested approaches are efficient and simple to use for solving fractional-order differential equations. The approximate and exact solutions of the partial fractional differential equations for integer order were compared. Additionally, the fractional-order and integer-order results are contrasted using simple tables. It has been confirmed that the solution produced using the provided methods converges to the exact solution at the appropriate rate. The primary advantage of the suggested method is the small number of computations needed. Moreover, it may be used to address fractional-order physical problems in a number of fields.

Application of Perturbation Method to Approximate the Solutions of Differential Equation

We investigate the core features of the perturbation method with the help of some simple but sophisticated problems and demonstrate how much accurately it predicts the solutions of the problems. To fulfill the target, we use the method for getting the solution of differential equations with initial and boundary conditions. Then the results obtained are compared with the series solution and the exact/numerical solution by using Mathematica and Fortran Programming. The comparisons are shown graphically. Also, the perturbation series approximation and the exact or numerical solution are in good agreement. Our investigation shows that a certain number of terms of the perturbation series gives an excellent approximation than the same number of terms of the numerical solution.

An innovative approach to approximating solutions of fractional partial differential equations

Abstract The RPS-M (residual power series method) is a valuable technique for solving F-PDEs (fractional partial differential equations). However, the derivative of the residual function to obtain the coefficients of the series is required in RPS-M. This makes the application of the classical RPS-M limited to a certain extent due to the complexity of the derivation of the residual function for higher iterations. To overcome this obstacle, in this study, we present a simplified version of this approach with the help of Laplace transform that requires less computation and offers higher accuracy. This modified method does not require derivation as well as limit of the residual function to estimate the unknown coefficients of the series solution. To demonstrate its effectiveness, we apply the proposed method to nonlinear F-PDEs to obtain their semi-analytical solution. The obtained solutions exhibit excellent agreement when compared to results obtained using other established approaches. We have also provided the convergence analysis of the obtained solution. Furthermore, by comparing the outcomes for various values of the non-integer order \(\sigma\), we observe that as the value approaches an integer order, the solution converges towards the exact solution.

Modified special functions: properties, integral transforms and applications to fractional differential equations

In this paper, we defined the modified gamma and beta functions, which involving the generalized M-series at their kernels, and then defined the modified Gauss and confluent hypergeometric functions by using the modified beta function. Furthermore, we presented some of their properties such that, integral representations, summation formulas and derivative formulas. Also, we applied beta, Mellin, Laplace, Sumudu, Elzaki and general integral transforms to newly defined the modified special functions. Then, we obtained solution of fractional differential equations involving the modified special functions, as examples. Finally, we gave the relationships between the modified functions with some of the generalized special functions, which can be found in the literature.

Application of shifted Vieta-Lucas polynomials for the numerical treatment of Volterra-integro differential equations

In this study, the numerical solution of the Volterra-integro differential equations was obtained by applying the variational iteration strategy with the shifted Vieta-Lucas polynomials. The proposed method builds the shifted Vieta-Lucas polynomials for the Volterra-integro differential equation which are then used as basis functions for the approximation. Numerical examples were given to establish the effectiveness and dependability of the recommended approach. Calculations were performed using Maple 2022 software.

Automated intelligent design of modified uni-traveling carrier photodectors

This paper introduces an automatic intelligent design method for the modified uni-traveling carrier photodetector (MUTC-PD). The conventional photodetector design process often relies on the numerical solution of complex nonlinear partial differential equations to simulate and optimize device performance, which is not only computationally intensive but also inefficient. To overcome this challenge, we apply the charge control principle to calculate the photodetector bandwidth, which improves the computational speed by a factor of approximately 1800 compared to the numerical solution of nonlinear partial differential equations. To further optimize the structure of the photodetector, we incorporate the Velocity Varying Climbing Particle Swarm Optimization (VVCPSO) algorithm. This is an improved algorithm based on the traditional particle swarm algorithm, which is able to quickly find the optimal solution in a complex parameter space. By applying the VVCPSO algorithm, we successfully fine-tuned the photodetector structure and obtained structural parameters with optimal performance. Our thorough verification process confirms that the proposed method is consistent with the results of ATLAS simulation software. Automated design has resulted in a high-performance MUTC-PD with a responsivity of 0.52A/W and a bandwidth of 60 GHz (@-3 V) at a mesa diameter of 16µm. Compared to the pre-optimized device, the bandwidth is increased to three times the original. By reducing the mesa diameter to 4µm, the bandwidth can be further increased to 82 GHz (@-3 V). The proposed method's calculation speed is fast enough, enabling extensive parameter studies to optimize device performance.

Analytical solution to boundary layer flow and convective heat transfer for low Prandtl number fluids under the magnetic field effect over a flat plate

AbstractThe present study aims to quantify the flow field, flow velocity, and heat transfer features over a horizontal flat plate under the influence of an applied magnetic field, with a particular emphasis on low Prandtl number fluids. Nonlinear partial differential expressions can be incorporated into the ordinary differential framework with the use of appropriate transformations. This research utilizes the variational iteration method (VIM) to approximate solutions for the system of nonlinear differential equations that define the problem. The objective is to demonstrate superior flexibility and broader application of the VIM in addressing heat transfer issues, compared to alternative approaches. The results obtained from the VIM are compared with numerical solutions, revealing a significant level of accuracy in the approximation. The numerical findings strongly suggest that the VIM is effective in providing precise numerical solutions for nonlinear differential equations. The analysis includes an examination of the flow field, velocity, and temperature distribution across various parameters. The study found that improving temperature patterns, velocity distribution, and flow dynamics were all positively impacted by increasing the Prandtl numbers. As a result, this leads to the thickness of the boundary layer to decrease and improves heat transfer at the moving surface. Thus, the convection process becomes more efficient. When the strength of the magnetic field is increased, the velocity of the fluid decreases. This observation aligns with expectations since the magnetic field hampers the natural flow of convection. Notably, the convection process can be precisely controlled by carefully applying magnetic force.

Analyzing the time-fractional (3 + 1)-dimensional nonlinear Schrödinger equation: a new Kudryashov approach and optical solutions

The paper focuses on investigating the time-fractional (3 + 1)-dimensional cubic and quantic nonlinear Schrödinger equation. We adopt the novel Kudryashov method to generate a distinct class of optical solutions for the current conformable fractional derivative problem. Our method explores various solution forms, including dark, wave, mixed dark-bright, and singular solutions. The soliton solutions we construct are visually represented to illustrate the influence of the fractional order derivative. Further, we elucidate the influence of solution parameters on the wave envelope, providing clear interpretations through 2D graphics presentations. The results underscore the efficacy of our approach in discovering exact solutions for nonlinear partial differential equations, especially in cases where alternative methods prove ineffective. The significance of the present paper lies in its contribution to advancing the understanding of the behavior of optical solutions in nonlinear systems, providing valuable insights for both theoretical and practical applications. In the field of nonlinear optics, this equation can describe the propagation of optical pulses in nonlinear media. It helps in understanding the behavior of intense laser beams as they propagate through materials exhibiting nonlinear optical effects such as self-focusing, self-phase modulation, and optical solitons.

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.

Existence and exponential stability of a periodic solution of an infinite delay differential system with applications to Cohen–Grossberg neural networks

In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincidence degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen–Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and two numerical examples to illustrate the effectiveness of some of our 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.

Reliable optimal controls for SEIR models in epidemiology

We present and compare two different optimal control approaches applied to SEIR models in epidemiology, which allow us to obtain some policies for controlling the spread of an epidemic. The first approach uses Dynamic Programming to characterise the value function of the problem as the solution of a partial differential equation, the Hamilton–Jacobi–Bellman equation, and derive the optimal policy in feedback form. The second is based on Pontryagin’s maximum principle and directly gives open-loop controls, via the solution of an optimality system of ordinary differential equations. This method, however, may not converge to the optimal solution. We propose a combination of the two methods in order to obtain high-quality and reliable solutions. Several simulations are presented and discussed, also checking first and second order necessary optimality conditions for the corresponding numerical solutions.

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.

The data-driven rogue waves of the Hirota equation by using Mix-training PINNs approach

Recently, solving differential equations by deep neural networks has attracted lots of attention, however, when the solution of differential equations has localized areas such as the rogue waves, the classical physics-informed neural networks (PINNs) cannot guarantee its prediction accuracy. In this paper, we propose two neural networks methods: mix-training physics-informed neural networks (MTPINNs) and prior information mix-training physics-informed neural networks (PMTPINNs), two deep learning models with more approximation ability based on PINNs. A series of numerical experiments show that the learning efficiency and absolute error accuracy of our proposed model can be improved significantly. The PMTPINNs model can so as to make up for the shortcomings of PINNs in this aspect. Furthermore, by testing the robustness of these two methods, for two kinds of rogue waves of the Hirota equation, it is surprising that these models also have good performance and thereby may provide more possibilities for solving mathematical physics problems better.

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.