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The conventional sonic boom propagation prediction method widely adopted in supersonic aircraft design involves a two-step procedure. In the first step, a compressible viscous or inviscid Computational Fluid Dynamics analysis is applied to the aircraft geometry at flight conditions to produce a flow field solution near the aircraft. Then in the second step, a one dimensional nonlinear Burgers' equation model is used to propagate sonic boom signature traces from the near field solution at flight altitude to the ground along a ray path. For an accurate ground signature prediction the near field signature must be accurately modeled at a sufficient distance from the aircraft flight path in order to minimize errors in the one dimensional propagation model. This is a very challenging task for general purpose CFD tools in a design environment because the cost of maintaining highly accurate off-body solutions increases dramatically as the radial distance is enlarged in the computational domain. It is also particularly difficult to apply these tools for wave propagation because the algorithms are normally lower order and numerically dissipative and dispersive. In this work a space marching procedure based on an optimized higher order finite difference method is developed and applied in conjunction with a CFD solution concentrated in the close vicinity of the aircraft. This new approach is much more efficient, compared to previous methods, in providing highly accurate near field signatures for full carpet ground predictions. Results indicate that near field signatures retain more waveform shape information farther from the aircraft geometry while reducing the CFD cost significantly. The predicted ground signature is also shown to converge in shape as the radial distance of the near field signature grows, which is indicative of a more ideal initial condition being supplied to the one dimensional wave propagation to ground. This feature is very difficult to replicate in a tractable manner with common CFD approaches used in design. [ABSTRACT FROM AUTHOR]

In this work, we study the optimal control of stochastic Burgers equation perturbed by Gaussian and Lévy-type noises with distributed control process acting on the state equation. We use dynamic programming approach for the feedback synthesis to obtain an infinite-dimensional second-order Hamilton–Jacobi–Bellman (HJB) equation consisting of an integro-differential operator with Lévy measure associated with the stochastic control problem. Using the regularizing properties of the transition semigroup corresponding to the stochastic Burgers equation and compactness arguments, we solve the HJB equation and the resultant feedback control problem. [ABSTRACT FROM AUTHOR]

In this work, the (2+1)-dimensional Painlevé integrable Burgers equation is investigated. By applying a certain unified method, some analytical solutions, involving rational functions, trigonometric functions and hyperbolic functions, are achieved. In order to predict the wave dynamics, several three-dimensional and two-dimensional graphs and contour profiles are constructed. Bright, dark, periodic, kink, anti-kink, singular, singular periodic, bell-shaped waves are thus obtained. The dynamics of these solutions can be illustrated graphically by choosing appropriate values for the parameters involved. Due to the presence of arbitrary constants in these derived solutions, they can be used to explain a variety of qualitative traits present in wave phenomena. The approach is efficient to algebraic computation and it can be used to categorize a wide range of wave forms, as shown by the demonstrated soliton solutions. Travelling wave solutions are converted into solitary wave solutions when certain values are set for the parameters. Using the Wolfram program Mathematica, we sketch the figures for various values of the associated parameters in order to closely examine the obtained solitons. [ABSTRACT FROM AUTHOR]

When dealing with the discretization of differential equations on non‐rectangular domains, a careful treatment of the boundary is mandatory and may result in implementation difficulties and in coefficient matrices without a prescribed structure. Here we examine the numerical solution of a two‐dimensional constant coefficient distributed‐order space‐fractional diffusion equation with a nonlinear term on a convex domain. To avoid the aforementioned inconvenience, we resort to the volume‐penalization method, which consists of embedding the domain into a rectangle and in adding a reaction penalization term to the original equation that dominates in the region outside the original domain and annihilates the solution correspondingly. Thanks to the volume‐penalization, methods designed for problems in rectangular domains are available for those in convex domains and by applying an implicit finite difference scheme we obtain coefficient matrices with a 2‐level Toeplitz structure plus a diagonal matrix which arises from the penalty term. As a consequence of the latter, we can describe the asymptotic eigenvalue distribution as the matrix size diverges as well as estimate the intrinsic asymptotic ill‐conditioning of the involved matrices. On these bases, we discuss the performances of the conjugate gradient with circulant and τ$$ \tau $$‐preconditioners and of the generalized minimal residual with split circulant and τ$$ \tau $$‐preconditioners and conduct related numerical experiments. [ABSTRACT FROM AUTHOR]

Recently, fractional derivatives have become increasingly important for describing phenomena occurring in science and engineering fields. In this paper, we consider a numerical method for solving the fractional Burgers' equations (FBEs), a vital topic in fractional partial differential equations. Due to the difficulty of the fractional derivatives, the nonlinear FBEs are linearized through the Rubin–Graves linearization scheme combined with the implicit the third-order Adams–Moulton scheme. Additionally, in the spatial direction of the FBEs, the fourth-order central finite difference scheme is used to obtain more accurate solutions. The convergence of the proposed scheme is theoretically and numerically analyzed. Also, the efficiency is demonstrated through several numerical experiments and compared with that of existing methods. [ABSTRACT FROM AUTHOR]

The discovery of partial differential equations (PDEs) from datasets has attracted increased attention. However, the discovery of governing equations from sparse data with high noise is still very challenging due to the difficulty of derivatives computation and the disturbance of noise. Moreover, the selection principles for the candidate library to meet physical laws need to be further studied. The invariance is one of the fundamental laws for governing equations. In this study, we propose an invariance constrained deep learning network (ICNet) for the discovery of PDEs. Considering that temporal and spatial translation invariance (Galilean invariance) is a fundamental property of physical laws, we filter the candidates that cannot meet the requirement of the Galilean transformations. Subsequently, we embedded the fixed and possible terms into the loss function of neural network, significantly countering the effect of sparse data with high noise. Then, by filtering out redundant terms without fixing learnable parameters during the training process, the governing equations discovered by the ICNet method can effectively approximate the real governing equations. We select the one-dimensional (1D) Kuramoto–Sivashinsky equation, the two-dimensional (2D) Burgers equation, the equation of two-dimensional channel flow over an obstacle, and the equation of three-dimensional (3D) intracranial aneurysm as examples to verify the superiority of the ICNet for fluid mechanics. Furthermore, we extend similar invariance methods to the discovery of wave equation (Lorentz Invariance) and verify it through single and coupled Klein–Gordon equation. The results show that the ICNet method with physical constraints exhibits excellent performance in governing equations discovery from sparse and noisy data. [ABSTRACT FROM AUTHOR]

A new class of vortices has been observed in the stratosphere following extreme wildfires (Canada 2017, Australia 2020) and volcanic eruptions (Raikoke 2019). These vortices are long‐lived mesoscale anticyclones (hundreds to 1,000 km in diameter) trapping plumes of aerosols and combustion/volcanic compounds. Owing to their unusual composition, these anticyclonically trapped plumes (ATPs) are associated with a significant radiative heating, which fuels their ascent through the stratosphere. This article investigates the dynamics of ATPs using two complementary approaches: analytically, in a potential vorticity (PV) perspective, and through idealised numerical simulations with the Weather Research and Forecasting (WRF) model. In both cases, we consider the vortical flow forced by a heating Lagrangian tracer. By reformulating the problem in the potential radius–potential temperature coordinate system introduced for tropical cyclones, we first clarify that ATP formation is concomitant with the injection of air into the stratosphere at extratropical latitudes. Then, we derive a set of simplified one‐dimensional equations describing the subsequent evolution of the flow after the injection. The equation obtained for the tracer is a variant of the classical Burgers' equation. In qualitative agreement with the three‐dimensional WRF simulations, this theoretical model predicts that ATPs develop an upper tracer front associated with sustained near‐zero anticyclonic PV, followed by a smooth tracer tail of cyclonic PV. Radiative relaxation of the temperature perturbations induced by the anticyclone and the presence of an initial PV anomaly tend to stabilise ATPs during their ascent. Finally, we note that the theory predicts a similar relationship between the plume and anticyclonic PV for cooled ATPs, which is supported by three‐dimensional simulations and may apply to the 2022 Hunga volcanic plume. [ABSTRACT FROM AUTHOR]

The time-periodic plane Poiseuille flow of electrically conducting incompressible generalized Burgers fluids through a porous medium is analytically and numerically investigated in the presence of a transverse uniform magnetic field. The main purpose is to provide analytical expressions for the dimensionless steady-state fluid velocity, non-trivial shear stress and Darcy's resistance, which can be used to bring to light important characteristics concerning fluid behavior. Similar solutions corresponding to the Poiseuille flow of the same fluids induced by a constant pressure gradient are obtained as limiting cases of previous results. The present results reduce to known solutions from the literature when magnetic and porous effects are neglected, and their validation is graphically proved. The needed time to reach a steady state has been graphically determined. It was found that the steady state is later obtained in the absence of a magnetic field and porous medium. The impact of a magnetic field and porous medium on the fluid velocity, shear stress and flow resistance has been systematically examined and elucidated through graphical representations. The findings reveal that the presence of a magnetic field or porous medium results in a reduction in the fluid velocity, accompanied by an increase in the flow resistance. [ABSTRACT FROM AUTHOR]

This research article introduces the four-dimensional natural transform Adomian decomposition method (FNADM) for solving the (3+1)-dimensional time-singular fractional coupled Burgers' equation, along with its associated initial conditions. The FNADM approach represents a fusion of four-dimensional natural transform techniques and Adomian decomposition methodologies. In order to observe the influence of time-Caputo fractional derivatives on the outcomes of the aforementioned models, two examples are illustrated along with their three-dimensional figures. The effectiveness and reliability of this approach are validated through the analysis of these examples related to the (3+1)-dimensional time-singular fractional coupled Burgers' equations. This study underscores the method's applicability and effectiveness in addressing the complex mathematical models encountered in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

This paper presents a study on the numerical solutions of the Burgers' equation with forcing effects. The article proposes three hybrid methods that combine two-point, three-point, and four-point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers' equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers' problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers' equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers-type parabolic partial differential equations arising in various fields of science and engineering. [ABSTRACT FROM AUTHOR]

In this work, we consider an inverse backward problem for a nonlinear parabolic equation of the Burgers' type with a memory term from final data. To this aim, we first establish the well-posedness of the direct problem. On the basis of the optimal control framework, the existence and necessary condition of the minimizer for the cost functional are established. The global uniqueness and stability of the minimizer are deduced from the necessary condition. Numerical experiments demonstrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

We study weighted inequalities of Hardy and Hardy–Poincaré type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile. [ABSTRACT FROM AUTHOR]

The iterated Crank-Nicolson (ICN) method is a successful numerical algorithm for solving partial differential equations. When unequal weights are used in the ICN predictor-corrector process, the convergence rate is reduced to first order. In this paper, we propose two modified ICN algorithms that achieve second order convergence rate, based on two different ways of choosing the weights. The first approach employs geometrically averaged weights in two consecutive iterations, and the second one uses arithmetically averaged weights for two consecutive time steps. The stability and second order accuracy of our methods are verified using stability and truncation error analysis and numerically demonstrated by solving linear and semi-linear hyperbolic partial differential equations, Burgers' equation, and the Gross-Pitaevskii equation. [ABSTRACT FROM AUTHOR]

The nonlinear evolution equations (NLEEs) play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences. The Drinfeld–Sokolov–Wilson (DSW) equation and the Burgers equation are the fundamental equations occurring in various areas of physics and applied mathematics, such as nonlinear acoustics and fluid mechanics. The new generalized (G ′ / G) -expansion method is an effective and more powerful mathematical tool for solving NLEEs arising in applied mathematics and mathematical physics. In this article, we investigate further exact solutions as well as soliton solutions to these couple of nonlinear evolution equations by executing the new generalized (G ′ / G) -expansion method. A large number of soliton solutions, including single soliton, bell-shaped soliton, kink-shaped soliton, singular kink soliton, singular soliton, periodic soliton, irregular periodic soliton solutions, and others, have been retrieved. Each of the derived solutions includes an explicit function of the variables in the equations under consideration. We provide some 3D plots to visualize and realize the characteristics of these solutions. It has been established that the suggested techniques are more potential and successful at obtaining soliton solutions for nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this study, with the help of improved exp (- Φ (η)) -function method, we explored the soliton solutions for the well-known nonlinear Fisher and nonlinear diffusion dynamical equations. Both nonlinear equations are useful for studying fluid dynamics, biology population models, plasma, fiber optics and crystallization. Our obtained results are in the form of singular bright solitons, combined bright-dark solitons, singular dark solitons, kink wave solitons, and anti-kink solitons as well as the physical structure of the extracted solutions as drawn by three-dim graphically by utilizing numerical simulation. The method is applied on two different nonlinear models, which reveal the use and simplicity of the method. It has been verified that the proposed technique yields a wide range of solutions and provides a more powerful computational approach for investigate the nonlinear evolution equations in engineering and the mathematical sciences. Our extracted results are interesting, more general, some novel and will play important role in various field of science such as nonlinear optics, quantum physics, nonlinear dynamics, laser optics, fiber optics, plasma physics, soliton wave theory, communicational system and field of engineering. The extracted results demonstrated that the utilized approach is effective, powerful, and efficient for soliton results to the nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This article aims to study the generalized exponential rational function method to solve the differential-difference Burgers' equation. Our approach is an efficient method for solving nonlinear partial differential equations, and it can be used for a specific type of nonlinear differential-difference equations. To recognize diverse singular soliton and multi-soliton wave structures, we displayed the 3-D and contour graphs associated with the solutions. [ABSTRACT FROM AUTHOR]

We study a stochastic Allen-Cahn equation with nonlinear diffusion and a mild multiplicative noise on a bounded domain in R n. We consider initial data which satisfy some non-degeneracy conditions, and prove that steep transition layers develop within a very short time. We then study the motion of these transition layers and derive a stochastic motion law for the sharp interface limit. [ABSTRACT FROM AUTHOR]

We study the dynamics of the following porous medium equation with strong absorption ∂ t u = Δ u m − | x | σ u q , posed for (t , x) ∈ (0 , ∞) × ℝ N , with m > 1 , q ∈ (0 , 1) and σ > 2 (1 − q) / (m − 1). Considering the Cauchy problem with non-negative initial condition u 0 ∈ L ∞ (ℝ N) , instantaneous shrinking and localization of supports for the solution u (t) at any t > 0 are established. With the help of this property, existence and uniqueness of a non-negative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new series of Legendre basis functions by using the matrix decomposition technique, which are simultaneously orthogonal in both L 2 - and H 1 -inner products. Then, we construct efficient space-time spectral method for linear Sobolev equation using spectral approximation in space and multi-domain collocation approximation in time, which can be implemented in a synchronous parallel fashion. Next, we propose a novel Legendre spectral element method for solving nonlinear Sobolev equation with Burgers' type term, which reduce the non-zero entries of linear systems and computational cost. Some rigorous error estimates are carried out for one-dimensional Sobolev equation. Numerical experiments illustrate the effectiveness and accuracy of the suggested approaches. [ABSTRACT FROM AUTHOR]

Accurate reporting and prediction of PM2.5 concentration is very important for improving public health. In this article, we use spectral clustering algorithm to cluster 15 cities in the Pearl River Delta. On this basis, we propose a special difference equation model, especially the use of nonlinear diffusion equations to characterize the temporal and spatial dynamic characteristics of PM2.5 propagation between and within clusters for real‐time prediction. For example, through the analysis of PM2.5 concentration data for 91 consecutive days in the Pearl River Delta, and according to different accuracy definitions, the average prediction accuracy of the difference equation model in all city clusters is 97% or 88%. The mean absolute error (MAE) of the forecast data for each urban agglomeration is within 7 units (μg/m3). Experimental results show that the difference equation model can effectively reduce the prediction time and improve the prediction accuracy. Therefore, based on the spectral clustering algorithm and the difference equation model, the fastest prediction speed and the best prediction result can be obtained, and the problem of PM2.5 concentration prediction can be effectively solved. The research can provide decision support for local air pollution early warning and urban integrated management. [ABSTRACT FROM AUTHOR]

The current paper concentrates on discovering the exact solutions of the time-fractional regular and singular coupled Burger's equations by involving a new technique known as the double Sumudu-generalized Laplace and Adomian decomposition method. Furthermore, some theorems of the double Sumudu-generalized Laplace properties are proved. Further, the offered method is a powerful tool for solving an enormous number of problems. The precision of the technique is evaluated with the aid of some examples, this method offers a solution precisely and successfully in a series form with smoothly calculated coefficients. The relation between both the approximate and exact solution is represented by a graph to display the high speed of this method's convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study three numerical schemes for the McKean-Vlasov equation {dXt=b(t,Xt,μt)dt+σ(t,Xt,μt)dBt,∀t∈[0,T],μt is the probability distribution of Xt, { d X t = b (t , X t , μ t) d t + σ (t , X t , μ t) d B t , ∀ t ∈ [ 0 , T ] , μ t  is the probability distribution of   X t , $ \{ \begin{array}{l}dX_t=b{(t,X_t,\mu_t)}dt+\sigma{(t,X_t,\mu_t)}dB_t,\\ \forall t\in\lbrack 0,T\rbrack,\mu_t\text{is the probability distribution of }X_t,\end{array} $ where X0 : (Ω, F, ℙ) → (ℝd, B(ℝd)) is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients b and σ, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends previous work [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] established in a one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme inspired by the K-means clustering). Two simulations are presented at the end of this paper: Burgers equation introduced in [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] and the network of FitzHugh- Nagumo neurons (see [J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, J. Math. Neurosci. 2 (2012) 1–50.] and [M. Bossy, O. Faugeras and D. Talay, J. Math. Neurosci. 5 (2015) 1–23.]) in dimension 3. [ABSTRACT FROM AUTHOR]

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article. [ABSTRACT FROM AUTHOR]

In this paper, we numerically solve the nonlinear time-fractional diffusion equation of distributed order on an unbounded domain with a weak singularity. A fully discrete implicit scheme is developed based on the L1 formula on graded meshes in time and the Galerkin spectral method using the Laguerre function in space. We obtained an α -robust discrete Gronwall inequality and the a priori error estimation of the numerical solution. Then, the existence and uniqueness of the numerical solution are discussed. Next, we present the α -robust stability and convergence of the fully discrete scheme, where the convergence was obtained based on the regularity conditions of the exact solution. A numerical example demonstrates the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by u (0 , t) − 2 , our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher's equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space. [ABSTRACT FROM AUTHOR]

The evolution of a turbulent flow to a statistically steady state can be cast as a multiscale problem involving energy redistribution processes that take place on the long, large eddy turnover timescale and chaotic processes that take place on the much shorter timescale of the turbulence fluctuations. But the absence of a way to perform super-resolution reconstructions of the instantaneous velocity field from its lower-dimensional moments has prevented the use of standard multiscale computational approaches for accelerating turbulence simulations. We introduce an encoder-decoder recurrent neural network model, an architecture typically used in natural language processing, for translating between the instantaneous velocity field and energy spectrum in a one-dimensional turbulent flow. We use the model in a multiscale simulation scheme to evolve the stochastic Burgers' equation and recover the final, statistically stationary turbulent Burgers' velocity field up to 443 times faster in wall-clock time than using direct numerical simulation alone. [ABSTRACT FROM AUTHOR]

In this paper, new symmetry reductions and similarity solutions for Burgers equation with moving boundary are obtained by means of Lie's method of infinitesimal transformation groups, for a linearly moving boundary as well as a parabolically moving boundary. By using discrete symmetries, new analytical solutions for the problem under consideration are presented, for two cases of the moving boundary: one moving with constant velocity and another one rapidly oscillating. [ABSTRACT FROM AUTHOR]

The one‐dimensional dynamics of nonlinear electrostatic dust‐ion acoustic waves (DIAWs), in a plasma containing Kaniadakis‐distributed electrons, fluid ions, and variable‐charge dust grains has been investigated. The κ−generalized electron charging current is derived based on the orbit‐motion limited approach. The reductive perturbation technique has been used to derive a KdV Burger equation of propagation. We have highlighted the dependence between the values of the dust particles charge Qd0 and the validity domain of the Kaniadakis parameter κ, where the Burger term C has positive. By integrating the KdV Burger equation numerically and assuming a laboratory dusty plasma, a transition from oscillatory to monotonic shock waves is observed with increasing values of κ. We have also found that the phase velocity of DIA shock waves decreases, and the dissipation can prevail over that of dispersion as the electrons move further away from thermal equilibrium (κ is increased). The coexistence of compressive and rarefactive of monotonic shock profile is observed. It is found that an increase in κ leads to an increase of the compressive monotonic shock wave and decrease the rarefactive monotonic shock wave. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear boundary diffusion equation of porous medium type arising from a boundary control problem. We give a complete and sharp characterization of the asymptotic behavior of its solutions, and prove the stability of its separable solutions. [ABSTRACT FROM AUTHOR]

This article presents an efficient implicit spline-based numerical technique to solve the time-fractional generalized coupled Burgers' equation. The time-fractional derivative is considered in the Caputo sense. The time discretization of the fractional derivative is discussed using the quadrature formula. The quasilinearization process is used to linearize this non-linear problem. In this work, the formulation of the numerical scheme is broadly discussed using cubic B-spline functions. The stability of the proposed method is proved theoretically through Von-Neumann analysis. The reliability and efficiency are demonstrated by numerical experiments that validate theoretical results via tables and plots. [ABSTRACT FROM AUTHOR]

A qualitative analysis of the electron-acoustic wave is taken in a collisional plasma having two-temperature electrons with a fixed ion background, where hot electrons follow kappa distribution. The collision between stationary ions and cold electrons is considered. Using reduced perturbation technique, the Burgers equation for the plasma system is derived. Using traveling wave transformation, we obtain the dynamical system corresponding to the plasma system. Phase plane analysis is used in the dynamical system to study different kinds of wave features for the considered plasma system. Moreover periodic wave features and shock wave features are investigated in accordance to periodic orbits and heteroclinic orbits obtained in the phase portrait. Role of the superthermal parameter (κ ), speed of the travelling wave (U) and α = n c 0 / n h 0 (where n c 0 denotes the number density of cold electrons in equilibrium and n h 0 denotes the number density of hot electrons in equilibrium) are shown on the electron-acoustic periodic waves and shock waves structures. The results hold relevance and significance in the context of space plasma. [ABSTRACT FROM AUTHOR]

This paper centers on constructing a lucid and utilitarian approach to tackle linear and non-linear two-dimensional partial differential equations. To test the applicability of the proposed algorithm a variant of the classical two-dimensional unsteady Burgers' equation is set up as a testing ground. The method in a nutshell reduces to solving one-dimensional Burgers' equations resulting from the application of appropriate operator splitting techniques in the temporal direction. In solving these one-dimensional Burgers' equations a refined tailored finite point method in conjunction with an apposite linearization to the purpose is employed. The conditional stability, consistency, and convergence of the method are established theoretically and the method is found to be first-order convergent in time and second-order convergent in space. To illustrate the accuracy of the scheme, divers examples have been solved and the results obtained prove that this method is top-notch in terms of cost-cutting and time efficiency through the sufficiency of coarse meshes. • This work presents a minimal machinery algorithm on an operator splitting technique in conjunction with a tailored FPM for a 2D Burgers' equation. • The tailored FPM is advisable for its efficient adaptation to local solution properties, user-friendly algorithm, and cost-effectiveness. • The manuscript offers a comprehensive mathematical and computational background along with stability, consistency, and convergence analyses. • The algorithm despite being simple and straightforward has performed on a par with other methods based on elite classical formulation techniques. [ABSTRACT FROM AUTHOR]

During recombination, the cosmic background radiation is disturbed, in particular, by Lyman-alpha emissions from neutral hydrogen. It is proposed to account for the subsequent time-dependent partial thermalization of the Ly α energy content in an analytically solvable nonlinear diffusion model. The amplitude of the partially thermalized and redshifted Ly α line is found to be too low to be visible in the cosmic microwave spectrum, in accordance with previous numerical models and Planck observations. [ABSTRACT FROM AUTHOR]

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α -model, convective Brinkman-Forchheimer equations, fractional (s , p) -Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others. [ABSTRACT FROM AUTHOR]

This work is a review of a group of results on the stochastic Burgers equation with small viscosity, obtained during the last two decades. These results jointly show that the equation makes a good model of hydrodynamical turbulence. The model provides natural and rigorously justified analogies of a number of key predictions of the theory of turbulence, including the main assertions of the Kolmogorov approach to turbulence, known as the K41 theory. The presentation is based on the recent book by Boritchev and Kuksin, One-Dimensional Turbulence and the Stochastic Burgers Equation (AMS, 2021) and on two papers that develop the book's results. Part of the assertions that we review here together with their rigorous proofs were earlier heuristically proved in works of physicists. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic Burgers' equation, which is forced by multiplicative noise in the Stratonovich sense. To get a new trigonometric and hyperbolic stochastic solutions, we apply $ \exp (-\varphi (\mu)) $ exp ⁡ (− φ (μ)) -expansion method. In addition, we demonstrate the effect of multiplicative noise on the exact solutions of the Burgers' equation by introducing some graphic representations. [ABSTRACT FROM AUTHOR]

We solve the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives for systems exhibiting noninteger power laws in their Hamiltonians. Based on the fractional Liouville equation, we calculate the density function (DF) of a classical ideal gas. If the Riemann–Liouville derivative is used, the DF is a function depending on both the momentum and the coordinate , but if the derivative in the Caputo sense is used, the DF is a constant independent of and . We also study a gas consisting of fractional oscillators in one-dimensional space and obtain that the DF of the system depends on the type of the derivative. [ABSTRACT FROM AUTHOR]

The second order Burger's equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative is addressed using Caputo's formula, while the CBS technique with the help of a θ -weighted scheme is utilized to discretize the first- and second-order spatial derivatives. The quasi-linearization technique is used to linearize the non-linear terms. The suggested scheme demonstrates unconditionally stable. Some numerical tests are utilized to evaluate the accuracy and feasibility of the current technique. [ABSTRACT FROM AUTHOR]

In applicative biosensor technology, mathematical modeling plays an indispensable role in explaining the transport of electrical signals by analyzing the binding behavior of the biochemical enzyme inhibitors to the target molecule. The biosensors are extensively used in clinical diagnostics, drug detention, food analysis and environment monitoring because they are highly sensitive, reliable and relatively cheap. Dynamic mathematical models used for biological investigation serve the purpose efficiently with very reasonable outcomes. In this study, a time-independent mathematical model for biosensor enzyme–substrate–inhibitor system under uncompetitive inhibition based on the nonlinear diffusion equations taking into consideration the kinetic rate constants and the initial concentrations of enzyme, substrate and inhibitor has been formulated and solved analytically using variational iteration method (VIM). The reliability and accuracy has been proved by comparing our results with the solution obtained by standard VIM. Chosen biosensors showed desirable sensitivity, selectivity and potential for application on real samples. They are frequently made to prevent interference from undesirable components that are present in the monitored system. The VIM is effectively and easily used to obtain solution of nonlinear equations accurately. Further, the solution has been discussed exhaustively for different values of reaction parameters avoiding linearization and unrealistic assumptions and the results obtained significantly agree with existing literature. [ABSTRACT FROM AUTHOR]

3D Burgers equation is an important model for turbulence. It is natural to expect the long-time behaviour for this hydrodynamics equation. However, there is no result about the long-time behaviour for this deterministic model. Surprisingly, if the system is perturbed by stochastic noise, we establish the existence and uniqueness of invariant measure for 3D stochastic Burgers equation. [ABSTRACT FROM AUTHOR]

A new strategy is presented for the construction of high-order spatial discretisations extracted from a lower-order basic discretisation. The key consideration is that any spatial discretisation of a derivative of a solution can be expressed as the exact differentiation of a corresponding 'filtered' solution. Hence, each numerical discretisation method may be directly linked to a unique spatial filter, expressing the truncation error of the basic method. By approximately deconvolving the implied filter of the basic numerical discretisation an augmented high-order method can be obtained. In fact, adopting a deconvolution of the implied filter of suitable higher order enables the formulation of a new spatial discretisation method of correspondingly higher order. This construction is illustrated for finite difference (FD) discretisation schemes, solving partial differential equations in fluid mechanics. Knowing the implied filter of the basic discretisation, one can derive a corresponding higher order method by approximately eliminating the implied spatial filter to a certain desired order. We use deconvolution to compensate for the implied filter. This corresponds to a 'sharpening' of numerical solution features before the application of the basic FD method. The combination will be referred to as Approximate Deconvolution Discretisation (ADD). The accuracy of the deconvolved FD scheme depends on the order of approximation of the deconvolution filter. We present the 'sharpening' of several well-known FD operators for first- and second-order derivatives and quantify the achieved accuracy in terms of the modified wavenumber spectrum. Examples include high-order extensions up to new schemes with spectral accuracy. The practicality of the deconvolved FD schemes is illustrated in various ways: (i) by investigation of exactly solvable advection and diffusion problems, (ii) by tracking the evolution of the numerical solution to the Taylor-Green vortex problem and (iii) by showing that ADD yields spectral accuracy for the Burgers equation and for double-jet flow of an incompressible fluid. [ABSTRACT FROM AUTHOR]

Partial differential equations have recently garnered substantial attention as an image processing framework due to their extensibility, the ability to rigorously engineer and analyse the governing dynamics as well as the ease of implementation using numerical methods. This paper explores a novel approach to image trinarization with a concrete real-world application of classifying regions of sperm images used in the automatic analysis of sperm morphology. The proposed methodology engineers a diffusion equation with non-linear source term, exhibiting three steady-states. The model is implemented as an image processor using a standard finite difference method to illustrate the efficacy of the proposed approach. The performance of the proposed approach is benchmarked against standard image clustering/segmentation methods and shown to be highly effective. • A novel approach to image trinarization using a dynamic PDE based framework. • Data driven Medical Imaging application (Sperm Morphology). • Provably stable method. [ABSTRACT FROM AUTHOR]

One-dimensional flows of an incompressible viscoelastic polymer fluid that are qualitatively similar to the solutions of Burgers' equation are described on the basis of mesoscopic approach for the first time. The corresponding initial boundary-value problem is posed for the system of quasilinear differential equations. The numerical algorithm for solving it is designed and verified. The algorithm uses the explicit fifth-order scheme to approximate unknown functions with respect to time variable and the rational barycentric interpolations with respect to space variable. A method for localization of singular points of the solution in the complex plain and for adaptation of the spatial grid to them is implemented using the Chebyshev-Padé approximations. Two regimes of evolution of the solution to the problem are discovered and characterized while using the algorithm: regime 1—a smooth solution exists in a sufficiently large time interval (the singular point moves parallel to the real axis in the complex plane); regime 2—the smooth solution blows up at the beginning of evolution (the singular point reaches the segment of the real axis where the problem is posed). We study the influence of the rheological parameters of fluid on the realizability of these regimes and on the length of time interval where the smooth solution exists. The obtained results are important for the analysis of laminar-turbulent transitions in viscoelastic polymer continua. [ABSTRACT FROM AUTHOR]

We study the nondiffusive Westervelt equation in the weakly nonlinear regime and show that the leading profile equation is of Burgers' type. We show that a compactly supported nonlinearity coefficientα can be reconstructed from the tilt of the transmitted high frequency wave packets sent from different directions, by demonstrating that those tilts are proportional to the X-ray transform ofα. [ABSTRACT FROM AUTHOR]

Motivated by porous medium equations with a randomly perturbed velocity field, this paper considers a class of nonlinear degenerate diffusion equations with nonlinear conservative noise in bounded domains. The existence, uniqueness, and L1-stability of nonnegative entropy solutions under the homogeneous Dirichlet boundary condition are proved. The approach combines Kruzhkov's doubling variables technique with a revised strong entropy condition that is automatically satisfied by the solutions of approximate equations. [ABSTRACT FROM AUTHOR]