Solutions Of Differential Equations Research Articles

New periodic solutions and solitary wave solutions for the time-fractional differential equations

New periodic solutions and solitary wave solutions for the time-fractional differential equations

Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps.

Existence and Uniqueness Solutions of Multi-Term Delay Caputo Fractional Differential Equations

This study investigates a novel type of nonlocal boundary value problem with multipoint-integral boundaries and multi-term delay Caputo fractional differential equations (FDE). The provided problem is turned into an analogous fixed-point problem using fixed-point (F P) theory tools. Additionally, discussing about stability, in Ulam-Hyers-Rassias (UHR), Ulam-Hyers (UH), generalized Ulam-Hyers-Rassias (GUHR) and generalized Ulam-Hyers (GUH) stability, for finding the problem. Based on our obtained results we given some examples. As of our obtained results are very useful to multi-term caputo FDE related to hydrodynamics.

Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the Borel–Padé approximant and classical analysis to determine the analytic structure of the solution using the first few terms of its asymptotic series. Afterward, we build an approximant, consistent with the resurgent properties of the equation. The procedure gives an approximate expression for the Borel–Ecalle resummation of the solution useful for practical applications. Connections with other physical applications are also discussed.

Classification of the non-null electrovacuum solution of Einstein-Maxwell equations with three-parameter abelian group of motions

The classification of the Stackel spaces of the electrovacuum of the type (3.0) has been done. These spaces are invariant under the action of the three-parameter abelian group of motions and belong to the first type Bianchi spaces. In the case of a non-zero cosmological term, the metrics and potentials contain solutions of a nonlinear ordinary differential equation of the second order. When the cosmological term equals zero, the metrics and the components of the electromagnetic field tensor are expressed through elementary functions.Thus the classification of the electrovacuum stackel spaces of all types is completed and complete list of these spaces is constructed.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

A Variant of Accelerated Ramadan Group Adomian Decomposition Method for Numerical Solution of Fractional Riccati Differential Equations

A Variant of Accelerated Ramadan Group Adomian Decomposition Method for Numerical Solution of Fractional Riccati Differential Equations

Approximate and Exact Solutions of Some Nonlinear Differential Equations Using the Novel Coupling Approach in the Sense of Conformable Fractional Derivative

Several scientific fields utilize fractional nonlinear partial differential equations to model various phenomena. However, most of these equations lack exact solutions. Consequently, techniques for obtaining approximate solutions, which sometimes yield exact solutions, are essential. In this research, we develop a new approach by combining the homotopy perturbation method (HPM) and the conformable natural transform to solve the gas-dynamic equation (GDE), the Fokker-Planck equation (FPE), and the Swift-Hohenberg equation (SHE) in the context of conformable derivatives. The proposed approach is called the conformable natural homotopy perturbation method (CNHPM). This approach has the advantage of not requiring assumptions about significant or minor physical factors. Consequently, it eliminates some of the constraints associated with conventional perturbation methods and can solve both weak and highly nonlinear problems. We consider the absolute, relative, and residual errors numerically and graphically to assess the correctness of our approach. The results show that our approach serves as a suitable alternative to the approximate methods in the literature for solving fractional differential equations.

The use of R code for modeling and simulation of 2-D contaminant migration through a soil for the pulse source

<p>In this study, based on an existing analytical solution for the two-dimensional transport of contaminants in a saturated soil layer, for the pulse source, the R program code was developed. The simulation is used to obtain the profiles of contaminant concentration, for a steady groundwater velocity, as a function of distance from the source and time. The problem is analytically solved by leveraging the similarity between the Gaussian (normal) distribution and contaminant concentration distribution, the development of the analytical model (i.e., solution of partial differential equation) by using this similarity is explained step by step since it may be of interest to researchers. Contaminant propagation is modeled using R software, which helps to understand how contaminants migrate through a saturated soil layer. This approach aids in comprehending the mechanisms and spatial dynamics of contaminant dispersion, facilitating the prediction and management of soil and groundwater contamination. The provided R code can be altered for different parameters and time intervals.</p>

The use of R code for modeling and simulation of 2-D contaminant migration through a soil for the pulse source

<p>In this study, based on an existing analytical solution for the two-dimensional transport of contaminants in a saturated soil layer, for the pulse source, the R program code was developed. The simulation is used to obtain the profiles of contaminant concentration, for a steady groundwater velocity, as a function of distance from the source and time. The problem is analytically solved by leveraging the similarity between the Gaussian (normal) distribution and contaminant concentration distribution, the development of the analytical model (i.e., solution of partial differential equation) by using this similarity is explained step by step since it may be of interest to researchers. Contaminant propagation is modeled using R software, which helps to understand how contaminants migrate through a saturated soil layer. This approach aids in comprehending the mechanisms and spatial dynamics of contaminant dispersion, facilitating the prediction and management of soil and groundwater contamination. The provided R code can be altered for different parameters and time intervals.</p>

On the construction of the solution of mean field stochastic differential equations driven by G-Brownian motion via small delays

On the construction of the solution of mean field stochastic differential equations driven by G-Brownian motion via small delays

A Comparison of Interpolation and Least Squares Methods for the Approximate Solution of Differential Equation

A Comparison of Interpolation and Least Squares Methods for the Approximate Solution of Differential Equation

Gyrator potential operator and Lp -Sobolev spaces involving Gyrator transform

This paper presents the properties of Gyrator potential operator G μ l and related L p -Sobolev spaces associated with Gyrator transform (GT). The Schwartz-type space S Δ ( R 2 ) is introduced. Various properties of the kernel of GT is studied including its continuity. Pseudo-differential operators A α , q associated with GT is defined and derived it's continuity on S Δ ( R 2 ) . The Gyrator potential operator G μ l is defined as a pseudo-differential operator associated with a precise symbol and obtained certain fruitful properties. The operator G μ l is extended to a space of distributions. Another integral representation of A α , q is obtained and its L p ( R 2 ) -boundedness result is derived. The spaces H α m , p ( R 2 ) and H Δ m , p ( R 2 ) are defined. It is shown that the operator G μ l is an isometry of H Δ m , p ( R 2 ) . An L p -boundedness result for the operator G μ l is proved. We conclude the article by applying some of the results to show that the analytical solution of certain non-homogeneous partial differential equations belong to these spaces.

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces

An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces

Additional Conditions in Boundary Value Problems of Heat Conduction (Review)

A review of studies related to the use of additional boundary conditions (ADBs) and additional sought functions (ADFs) in obtaining analytical solutions to heat conduction problems is presented. ADВs allow the equation to be executed at the boundaries, which leads to its execution inside the domain, excluding direct integration over the spatial coordinate. ADF allows one to reduce a partial differential equation to an ordinary differential equation, from the solution of which the eigenvalues of the boundary value problem are found. Eigenvalues in classical methods are found from the solution of the Sturm–Liouville boundary value problem formulated in the domain of a spatial variable. Consequently, the method used in this work leads to another algorithm for their determination, based on the solution of a temporary differential equation, the order of which is determined by the number of approximations of the resulting solution. In a problem based on determining the front of a temperature disturbance, the equivalence of solutions to the parabolic and hyperbolic heat equations was found. And, in particular, a number of approximations have been found that limit the speed of propagation of a thermal wave in the solution of a parabolic equation to a value equal to its real value for a specific material, at which it coincides with the solution of the hyperbolic equation.

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Specifically, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary differential equation from one of the conservation laws. Multiple methods, for example, the dynamical systems method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary differential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related PDE systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These findings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.

Transient Reviving Dynamics with an Exact Solution for Delay Differential Equations

Transient Reviving Dynamics with an Exact Solution for Delay Differential Equations