Solution Of Equation Research Articles

Enhancing Accuracy and Efficiency in Stiff ODE Integration Using Variable Step Diagonal BBDF Approaches

Recent advancements in mathematical modelling have uncovered a growing number of systems exhibiting stiffness, a phenomenon that challenges the effectiveness of traditional numerical methods. Motivated by the need for more robust numerical techniques to address this issue, this paper presents an enhanced version of the Diagonally Block Backward Differentiation Formula (BBDF) that incorporates intermediate points, known as off-step points, to improve the accuracy and efficiency of solutions for stiff ordinary differential equations (ODEs). The new scheme leverages an adaptive step-size strategy to refine accuracy and efficiency between regular and off-grid integration steps. Theoretical analysis confirms that the proposed scheme is an A-stable and convergent method, as it satisfies the fundamental criteria of consistency, zero-stability, and A-stability. Numerical experiments on single and multivariable systems across varying time scales demonstrate significant improvements in solving stiff ODEs compared to existing techniques. Therefore, the new proposed method is an effective solver for stiff ODEs.

Fredholm Determinant and Wronskian Representations of the Solutions to the Schrödinger Equation with a KdV-Potential

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials.

Massless field equations for spin 3/2 in dimension 6

Main topic of the paper is a study of properties of massless fields of spin 3/2. A lot of information is available already for massless fields in dimension 4. Here, we concentrate on dimension 6 and we are using the fact that the group SL(4,C) is isomorphic with the group Spin(6,C). It makes it possible to use tensor formalism for massless fields. Main problems treated in the paper are a description of fields which need to be considered in the spin 3/2 case, a suitable choice of equations they should satisfy, irreducibility of homogeneous solutions of massless field equations, the Fischer decomposition and the Howe duality for such fields.

Electromagnetic Model of K‐Changes

AbstractK‐changes are observed as step‐like increases in the thundercloud electric fields. The K‐changes occur in the late part of intra‐cloud lightning or during negative cloud‐to‐ground lightning between return strokes. It has been shown that the processes leading to K‐changes initiate in the decayed part of a positive leader channel and propagate toward the flash origin. They are often accompanied by microsecond‐scale electric field pulses. We introduce a new model to simulate processes leading to the K‐changes in cloud‐to‐ground lightning. Our method is based on the full solution of Maxwell's equations coupled to Poisson's equation for the thundercloud charge structure. To model the K‐changes, we gradually increase the decayed channel conductivity. The modeled current wavefront propagates due to the K‐processes downward along a vertical channel and completely attenuates before reaching the ground. We derive the evolution of the linear charge densities and the scalar electric potential along the channel leading to K‐changes. We model electrostatic step‐like changes in the measured electric field together with the approximate rates and amplitudes of the microsecond scale pulses. Step‐like changes increase their amplitudes with the length of the simulated channel and with a higher conductivity of the channel. The microsecond‐scale pulse waveshapes depend mainly on the propagation velocity of the current wave, and the time scale of the conductivity increase. We show that our modeled waveforms are in a good agreement with observations conducted in Florida.

Robust and accurate numerical framework for multi-dimensional fractional-order telegraph equations using Jacobi/Jacobi-Romanovski spectral technique

This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs.

A novel efficient analytical technique and its application to fractional Cauchy reaction–diffusion equations

In this research paper, we suggest a novel analytical method to obtain the approximate solutions of linear and nonlinear differential equations of arbitrary order. The proposed technique is a good combination of Kharrat–Toma transform and q-homotopy analysis method. Additionally, we also find out the solution of Cauchy reaction–diffusion equation associated with regularized version of Hilfer–Prabhakar fractional derivative. The Cauchy reaction–diffusion equation is broadly used to describe the dynamical processes in physics, geology, biology, ecology, etc. Some characteristic properties of the considered model and existence as well as the uniqueness of the solutions are also discussed. The outcomes of the considered model are exhibited graphically to show the accuracy and efficiency of the suggested method.

Polynomial Analogs of Theta Functions and Rational Solutions of the KP Equation

Abstract In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.

Nonlinear analysis of compound pendulum model

Abstract This paper investigates the relationships among various nonlinear physi cal equations, with a particular focus on the standard form of the ϕ^4 equation. Based on the theoretical framework of the Jacobi elliptic function, an exact solution for the ϕ^4 equation is derived. A key innovation of this work is the discovery of the consistency between the ϕ^4 equation and the motion equation of the compound pendulum. By utilizing this correspondence, an exact solution for the compound pendulum equation is obtained, grounded in the Jacobi elliptic function theory. Compared to numerical methods, this solution provides higher accuracy and has the potential to be applied to
more complex nonlinear physical systems. This model can also be applied in areas such as vibration damping in building materials, mechanical system analysis, and spacecraft control.

Study of a non-linear Volterra integro-differential equation with a non-linear unknown source term

In this paper, we propose a new class of integro-differential nonlinear Volterra equations with a non-linear unknown source term. Our study focuses on both theoretical analysis and numerical approximation of solutions to these equations. In the theoretical analysis partition, we define the specific assumptions on the integral kernel's characteristics, on the source term function and also on the derivative of the solutions of equations, under which the Krasnoselskii's fixed point theorem can be applied and in order to ensures the existence and uniqueness of solutions to the proposed integro-differential equations. In the numerical approximation partition, we use a well-known numerical technique for solving integral equations called Nyström method. This method allows us to get an approximate solutions of the proposed equations. Furthermore, we provide some illustrative examples and according to the numerical method described in this paper we approach them, where the comparison results between the exact and approximate solutions of these examples confirm the effectiveness of the numerical Nyström method for approaching the solutions of this class of integro-differential equations.

A space-time meshfree method for heat transfer analysis in porous material

Abstract Porous material batteries are a new type of battery technology that uses porous materials as electrode materials, with advantages such as long lifespan. In the present study, we manily propose a space-time meshfree method for heat transfer problems in porous material energy storage battery. The thermal behavior is demonstrated based on a numerical solution of the energy conservation equation. A simple space-time meshless collocation scheme, which is based on a new type space-time radial basis function, is presented to get the approximate solution of the energy conservation equation. The energy conservation equation was transferred into a system of linear algebraic equations. By introducing prescribed boundary conditions, the heat transfer analysis in porous material energy storage battery can be shown with numerical results. Numerical discussions show that the proposed meshless collocation method is appropriate to simulate thermal behavior in porous material energy storage battery, while the traditional finite element method failed to provide the reasonable results for the tested examples in this paper.

Determining the constant diffusion coefficient by Fourier series solution of the diffusion equation

The purpose of this study was to obtain the diffusion coefficient by calculating directly from the Fourier series solution of the diffusion equation, which has not been studied so far. The Fourier series solution is the sum of a steady-state solution and a transient solution. The steady-state solution is a linear function of distance. The transient solution is the product of a distance function and a time function. The Fourier sine series solution, a function of distance only, is negative for the steady-state solution. For the sublimation diffusion of disperse dye in paste into polyethylene terephthalate (PET) film using the film-roll method, the constant surface concentration and the diffusion distance were determined from the quadratic regression curve for the concentration distribution at a long time. The concentration distribution of the transient solution is a downwardly convex curve with negative values. The concentration distribution of the Fourier series solution is also a downwardly convex curve. The diffusion coefficient in the exponential term, a function of time only, was determined from the value of the exponential term obtained by the surface concentration and the diffusion distance. The constant diffusion coefficients determined from the Fourier series solution are reliable because they are very similar to those obtained from the error function solution and have excellent linearity of the linear regression lines for their Arrhenius plots.

Anomalous Random Flights and Time-Fractional Run-and-Tumble Equations

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provide a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They display an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions.

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

In this paper, we obtain many different types of exact solutions to the time-fractional Klein–Gordon equation and the time-fractional generalized Hirota-Satsuma coupled KdV system by using the modified rational function approach. Some new solutions such as the kink-periodic solution, the anti-kink-periodic solution and the concave-convex-periodic solution are constructed. Furthermore, the kink and the singular kink waves, the bell shaped soliton and the singular soliton solutions of the two equations also are found. Some numerical simulations are presented, these works can effectively reflect the propagation phenomena of time-fractional nonlinear systems, and also enable us to understand time-fractional nonlinear physical phenomena more clearly.

First-principles studies on the electronic and thermal transport properties of twin graphene and its analogues

The novel two-dimensional (2D) carbon allotropes with various carbon networks have provided an unprecedented platform to explore fascinating device applications beyond graphene. In this work, the electronic and thermal transport properties of the twin graphene and its structural analogues, i.e., γ-graphyne, twin T-graphene, and twin 4–8 graphene, have been systematically revealed through first-principles calculations. Our results confirm the energetic and dynamical stability of the twin graphene family, and the intrinsic semiconducting nature of these 2D carbon sheets superior to graphene. Based on the solution of the phonon Boltzmann transport equation, the evaluated thermal conductivity of the considered 2D carbon sheets indicates that the absence of acetylenic linkages in carbon networks leads to a relatively enhanced heat transfer capacity, i.e., a higher thermal conductivity in the twin graphene family than the γ-graphyne case. More interestingly, a way to effectively tune thermal transport properties in the twin graphene family has been proposed via the utilization of atom-embedded carbon nanocages. Our results indicate that a notable 63.8% reduction in thermal conductivity can be achieved for twin graphene through the embedding of Ti atoms into the nanocages, exhibiting great potential for robust thermal management in low-dimensional carbon networks.

Solutions of differential equations in bicomplex space using Sadik transforms

Solutions of differential equations in bicomplex space using Sadik transforms

Computation of Deformable Interface Two‐Phase Flows: A Semi‐Lagrangian Finite Element Approach

ABSTRACTThis work aims at presenting a new computational approach to study two and three dimensional two‐phase flows and two dimensional coalescence phenomenon using direct numerical simulation. The flows are modeled by the incompressible Navier–Stokes equations, which are approximated by the finite element method. The Galerkin formulation is used to discretize the Navier–Stokes equations in the spatial domain and the semi‐Lagrangian method is used to discretize the material derivative. In order to satisfy the Ladyzhenskaya–Babuška–Brezzi condition, high‐order stable pair of elements are used, with pressure and velocity fields being calculated on different degrees of freedom in the unstructured mesh nodes. The interface is modeled by an unfitted adaptive moving mesh, where interface nodes are tracked in a Lagrangian fashion and moved with the velocity solution of the fluid motion equations. The surface tension is computed using the interface curvature and the gradient of a Heaviside function, and added in the momentum equations as a body force. In order to avoid undesired spurious modes at the interface due to high property ratios, a smooth transition between fluid properties is defined on the interface region. Several benchmark tests have been carried out to validate the proposed approach, and the obtained results have demonstrated agreement with analytical solutions and numerical results reported in the literature. A coalescence model is also proposed based on geometric criteria and results show interesting dynamics.

Novel nonlinear wave transitions and interactions for (2+1)-dimensional generalized fifth-order KdV equation

The (2 + 1)-dimensional generalized fifth-order KdV (2GKdV) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKdV equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped (XS/YS), asymmetric M-Shaped (MS), W-Shaped (WS), Space-Curved (SC) and Oscillation M-Shaped (OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.

Comparison of formulas derived from Fourier series solution of diffusion equation

The first Fourier series solution satisfying the boundary condition of zero concentration at the diffusion distance was derived by assuming that the solution consists of a steady-state solution and a transient solution. The first total amount which has passed through the first layer surface was derived from the first Fourier series solution. When the diffusion distance is large, the exponential term in the first total amount cannot be zero. For the sublimation diffusion of disperse dye in paste into polyethylene terephthalate film using the film-roll method, the plot of the first total amount against time was in good agreement with the quadratic regression curve. The second Fourier series solution satisfying the boundary condition of constant concentration at the diffusion distance was derived by assuming that the diffusion distance is the thickness of the first layer. The second total amount which has passed through the second layer surface was derived from the second Fourier series solution and is a linear function of time because the exponential term is zero due to the very small diffusion distance, the thickness of the first layer. The plot of the second total amount against time was in good agreement with the linear regression line. The time-lag formula was derived from the second total amount. The diffusion coefficients determined from the second total amount are much larger than those of the other three types. All four types showed excellent linearity in Arrhenius plot, but the activation energy for the time-lag formula is somewhat greater than the rest.

Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G′/G-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.

Novel results from quadratically nonlinear elastic wave models using Murnaghan’s potential

AbstractIn this article, we study one, two and three-dimensional nonlinear elastic wave equations using quadratically nonlinear Murnaghan potential. We employ two effective methods for obtaining approximate series solutions the Adomian decomposition and the variational iteration method. These methods have the advantage of not requiring any physical parametric assumptions in the problem. Finally, these methods can generate expansion solutions for linear and nonlinear differential equations without perturbation, linearization, or discretization. The results obtained using the adopted methods along various initial and boundary conditions are in excellent agreement with the numerical results on MATLAB, which show the reliability of our methods to these problems. We came to the conclusion that our methods are accurate and simple to use.