System Of Partial Differential Equations Research Articles

Analytical solutions and instability analysis of truncated M-fractional coupled dispersionless equations

Abstract This paper comprehensively investigates the truncated M-fractional coupled dispersionless equations, a nonlinear system of partial differential equations characterized by its M-fractional derivative. The Jacobi elliptic function expansion method is employed to derive analytical solutions for the coupled system. In addition, the modulation instability of the solutions is thoroughly explored, providing a detailed exposition of the mathematical framework governing the system. The analytical solutions are graphically illustrated and analyzed to highlight their physical significance. These findings have significant applications in nonlinear optics, offering new insights into wave propagation and stability within such systems.

Lie symmetry method to discuss the optimal system of Lie subalgebras and similarity solutions for shock propagation in a perfectly conducting dusty gas with magnetic field in rotating medium

In this paper, the similarity solutions for the shock wave propagation in a perfectly conducting dusty gas under the effect of axial or azimuthal magnetic field in the case of rotating medium by using Lie symmetry method have been discussed. The optimal system of one-dimensional Lie subalgebras related to the hyperbolic system of nonlinear partial differential equations had been constructed by using general invariant function and general adjoint transformation matrix. The optimal system is the collection of inequivalent optimal classes, which provides crucial information regarding different types of the similarity solutions. On the basis of the optimal system, we have discussed all possible similarity solutions for inequivalent optimal classes individually. Also, we have found that the similarity solutions exist in two cases for power law shock path and in one case with exponential law shock path. The shock wave decay is due to an increment in the physical parameter such as shock Cowling number or non-idealness parameter or adiabatic exponent or rotational parameter. For smaller value of the ratio of density of solid particles to initial species density of the gas i.e. [Formula: see text], the shock strength decreases with an increment in the mass fraction of solid particles in the mixture [Formula: see text], but the shock strength decreases as [Formula: see text] increases for [Formula: see text]. Also, the shock strength increases with an increment in [Formula: see text] at constant [Formula: see text]. It is observed that the shock wave is stronger in the presence of axial magnetic field as compared to azimuthal magnetic field.

Classical-quantum simulation of non-equilibrium Marshak waves

In the radiation hydrodynamic simulations used to design inertial confinement fusion (ICF) and pulsed power experiments, nonlinear radiation diffusion tends to dominate CPU time. This raises the interesting question of whether a quantum algorithm can be found for nonlinear radiation diffusion which provides a quantum speedup. Recently, such a quantum algorithm was introduced based on a quantum algorithm for solving systems of nonlinear partial differential equations (PDEs) which provides a quadratic quantum speedup. Here, we apply this quantum PDE (QPDE) algorithm to the problem of a non-equilibrium Marshak wave propagating through a cold, semi-infinite, optically thick target, where the radiation and matter fields are not assumed to be in local thermodynamic equilibrium. The dynamics is governed by a coupled pair of nonlinear PDEs which are solved using the QPDE algorithm, as well as two standard PDE solvers: (i) Python's py-pde solver; and (ii) the KULL ICF simulation code developed at Lawrence-Livermore National Laboratory. We compare the simulation results obtained using the QPDE algorithm and the standard PDE solvers and find excellent agreement.

PARASIAN OVER PARISIAN, HOW MUCH EARLIER SHOULD ONE EXERCISE?

In this paper, an integral equation approach used for pricing American-style Parisian options is extended to pricing Parasian options, after overcoming an additional difficulty of losing the “reset” feature of the latter in addition to still dealing with the high nonlinearity associated with American-style exercises. More specifically, compared with an American Parisian option, the accumulative feature of the “tracking clock” time results in a pair of coupled three-dimensional (3D) Partial Differential Equation (PDE) systems instead of coupled two-dimensional (2D) and 3D PDE systems. Upon successfully establishing the couple integral equation systems, we have successfully computed the option price and the optimal exercise price and compared the results with those of its Parisian counterpart. Through the comparison, we can directly observe how the accumulativeness of the “tracking clock” time affects the option and the optimal exercise prices and provide some meaningful financial explanations.

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

Well-balanced High-order Finite Difference Weighted Essentially Nonoscillatory Schemes for a First-order Z4 Formulation of the Einstein Field Equations

We develop a new class of high-order accurate well-balanced finite difference (FD) weighted essentially nonoscillatory (WENO) methods for numerical general relativity (GR), which can be applied to any first-order reduction of the Einstein field equations, even if nonconservative terms are present. We choose the first-order nonconservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. By introducing auxiliary variables, the vacuum Einstein equations in first-order form constitute a partial differential equation system of 54 equations that is naturally nonconservative. We show how to design FD-WENO schemes that can handle nonconservative products. Different variants of FD WENO are discussed, with an eye to their suitability for higher-order accurate formulations for numerical GR. We successfully solve a set of fundamental tests of numerical GR with up to ninth-order spatial accuracy. Due to their intrinsic robustness, flexibility, and ease of implementation, FD-WENO schemes can effectively replace traditional central finite differencing in any first-order formulation of the Einstein field equations, without any artificial viscosity. When used in combination with well-balancing, the new numerical schemes preserve stationary equilibrium solutions of the Einstein equations exactly. This is particularly relevant in view of the numerical study of the quasi-normal modes of oscillations of relevant astrophysical sources. In conclusion, general relativistic high-energy astrophysics could benefit from this new class of numerical schemes and the ecosystem of desirable capabilities built around them.

Employing the Sadik Residual Power Series Method to Analyze a System of Nonlinear Caputo Time-Fractional Partial Differential Equations

The present study proposes an approximate analytical solution to a nonlinear system of time-fractional partial differential equations. The Sadik residual power series method, which integrates the two-part Sadik integral transform with the residual power series technique, is employed to solve the fractional differential equation in the Caputo sense. Nonlinear problems with known and unknown solutions are examined to demonstrate the capacity of the technique. Numerical simulations and 3D visualizations are conducted for various values of the fractional order to further understand the solution’s behavior. Additionally, the results are validated against exact solutions or existing methodologies to ensure their reliability and accuracy. A key advantage of the proposed method is its ability to generate results without the need for Adomian polynomials, perturbation techniques, discretization, or linearization, enabling a more efficient.

Multidimensional backward stochastic differential equations with rough drifts

In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with an additional rough drift (rough BSDE), and give the existence and uniqueness of the adapted solution, either when the terminal value and the geometric rough path are small, or when each component of the rough drift only depends on the corresponding component of the first unknown variable (but we drop the one-dimensional assumption of Diehl and Friz [Ann. Probab. 40 (2012), pp. 1715–1758]). We also introduce a new notion of the p p -rough stochastic integral for p ∈ [ 2 , 3 ) p \in \left [2, 3\right ) , and then succeed in giving—through a fixed-point argument—a general existence and uniqueness result on a multidimensional rough BSDE with a general square-integrable terminal value, allowing the rough drift to be random and time-varying but having to be linear; furthermore, we connect it to a system of rough partial differential equations.

Enhanced Adomian Decomposition Method for Accurate Numerical Solutions of PDE Systems

This research addresses critical challenges in numerical solutions, which are vital for various engineering and physical fields. The Modified Adomian Decomposition Method (MADM) is proposed as a novel approach for solving linear and nonlinear partial differential equations (PDEs). MADM builds upon the Adomian Decomposition Method (ADM) by incorporating a new integral operator that significantly improves convergence rates and accuracy. Numerical examples demonstrate the effectiveness of MADM in handling complex nonlinear PDEs. Compared to traditional ADM, MADM consistently achieves more accurate and rapidly converging solutions. This enhancement is attributed to the novel integral operator, which addresses the limitations of ADM for intricate nonlinear problems. The paper outlines the application of MADM, its solution procedure, and its effectiveness through numerical examples. Comparisons with standard ADM solutions and exact solutions validate MADM's accuracy and superiority. The results suggest that MADM is a promising tool for expanding the applicability of Adomian methods in the field of solving PDEs.

Model-based workflow for sustainable production of high-quality spirits in packed column stills

This study addresses water scarcity in Chilean distilleries by developing a model-based engineering workflow. Prolonged droughts, likely driven by global warming, have intensified this problem. The primary challenge is maintaining high-quality spirits while reducing cooling water and energy use during batch distillation in packed columns. This process was modeled using mass and energy balances, resulting in a system of partial differential algebraic equations (PDAE). Our workflow includes mechanistic modeling, disturbance modeling, multiobjective optimization, model predictive control, and Monte Carlo simulations. Our findings show that cooling water consumption can be reduced by up to 35 % and energy consumption by up to 14.4 % while maintaining product quality. The proposed system is robust against operational disturbances and model mismatch, ensuring consistent distillate quality. This research demonstrates the integration of model-based optimization and control strategies in batch distillation processes, which can be replicated in other fruit wine distillation processes for improved sustainability.

From ABC to KPZ

AbstractWe study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with $$N\in {\mathbb {N}}$$ N ∈ N points, denoted by $${\mathbb {T}}_N$$ T N , and with three species of particles that we name A, B and C, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit $$N\rightarrow \infty $$ N → ∞ , to a system of stochastic partial differential equations, that can either be the Ornstein–Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann–Lebesgue lemma which is of independent interest.

Parallel two‐scale finite element implementation of a system with varying microstructure

AbstractWe propose a two‐scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed pullback operator, we are able to model the different microscopic domains as macroscopically dependent deformations of a reference domain. This allows for a relatively simple finite element framework to approximate the underlying system of partial differential equations with a parallel computational structure. We apply this technique to a model problem where we focus on transport in plant tissues. We illustrate the accuracy of the implementation with convergence benchmarks and show satisfactory parallelization speed‐ups. We further highlight the effect of the heterogeneous microscopic structure on the output of the two‐scale systems. Our implementation (publicly available on GitHub) builds on the deal.II FEM library. Application of this technique allows for an increased capacity of microscopic detail in multiscale modeling, while keeping running costs manageable.

Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies

Investigation of nanofluid flows under various conditions is becoming fascinating research due to its numerous applications in many science and engineering fields such as solar energy, geothermal energy, water purification, glass blowing, enhanced oil recovery, petroleum production and food processing. For the present requirements in thermal energy applications, this mathematical model is considered to explore the diffusion and heat source effects on the flow of conductive Casson and Maxwell nanofluids in a porous medium on an elongating sheet with a chemical reaction. Further, the effects of viscous dissipation and thermal radiation are also examined. Moreover, the entropy generation is analyzed since this approach is employed in solar energy exchangers. The governing equations and the corresponding boundary conditions are constructed. This system of partial differential equations (PDEs) is changed into a system of nonlinear ordinary differential equations (ODEs) with the help of a suitable transformation and then solved numerically by the bvp5c MATLAB package. The influences of the physical parameters on the flow phenomena are discussed and presented through figures and tables. This analysis explains that the Casson fluid flow has a higher velocity field than the Maxwell fluid flow. The Nusselt number becomes higher as the Dufour numbers increase. It has also been noticed that increasing the Brinkmann number improves the entropy and, an increase in radiation enriches the temperature profile. The concentration of the fluid is decreased for the increasing values of the higher-order chemical reaction parameter. Furthermore, comparisons of the current results were conducted for limiting examples of the problem and found to be in good agreement with existing results in the literature.

Symmetry Reductions of the (1 + 1)-Dimensional Broer–Kaup System Using the Generalized Double Reduction Method

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs.

Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions

Abstract Compressible Euler equations with space-dependent damping in high dimensions R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) are considered in this article. Assuming that the small initial velocity and small perturbation of the initial density have compact support, we establish finite-time blow-up results for the Euler system, by combining energy estimate and new test functions constructed by the solutions of the following linear elliptic partial differential equations system: − G 1 ( x ) + ∇ ⋅ G 2 → ( x ) = 0 , − G 2 → ( x ) + ∇ G 1 ( x ) = μ G 2 → ( x ) ( 1 + ∣ x ∣ ) λ . \left\{\begin{array}{l}-{G}_{1}\left(x)+\nabla \cdot \overrightarrow{{G}_{2}}\left(x)=0,\\ -\overrightarrow{{G}_{2}}\left(x)+\nabla {G}_{1}\left(x)=\frac{\mu \overrightarrow{{G}_{2}}\left(x)}{{(1+| x| )}^{\lambda }}.\end{array}\right. This result generalizes the one in the literature from 1 − D 1-D to high dimension R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) .

Heat transfer inspection and entropy optimization in water/MoS2-SiO2 Casson hybrid nanofluid flow with quadratic radiation ray over an exponentially contracting permeable Riga surface: Duality and stability analysis

In this study, we have examined the primary behaviour of a hybridized electro-magneto-non-Newtonian MoS2-SiO2/H2O nanofluid over a porous Riga surface that is exponentially contracting, in particular its flow dynamics and heat transfer characteristics. The key factors of this research are the presence of quadratic thermal radiation over the Riga surface under velocity slip and convective boundary. We have solved the highly circuitous coupled partial differential equations system by utilizing the bvp4c numerical method with MATLAB programming. A dual solution is determined based on a certain amount of mass suction parameter and shrinking parameter under the critical value range; beyond that value, no solution exists. We investigated the stability of the dual solutions by determining the minimal eigenvalue. An evaluation of the temporal stability of these solutions reveals that only the first solution remains stable. This is supported by the positivity of the smallest eigenvalues (β1> 0), indicating stability. Conversely, for the second solution, the smallest eigenvalues (β1< 0) are negative, signifying its instability. It can be seen that the nanofluid temperature is substantially enhanced for both branches of solution for the quadratic thermal radiating parameter, Eckert number, and Biot number. It was found that, in the first solution branch, the velocity layout was boosted with the velocity slip factor while reduced by enhancing the Casson parameter, whereas, in the second solution branch, the opposite behavior was observed. As the suction parameter increases, the heat transfer coefficient for the first solution is decreased by about 0.5 % and increases by about 0.75 % for the second solution branch. Quadratic thermal radiation parameter Nr significantly impacts the Nusselt number, as it is reduced by approximately 27.44 % and 28.72 % for the first and second solution branches, respectively. The shear stress coefficient is decreased by about 29.32 % for the first solution branch with contracting parameter λ. Because the Riga plate, paired with Casson nanofluid flows, has industrial and civil engineering applications, this study's thermal relevance makes it applicable to a wide range of industrial and engineering disciplines.

Model Order Reduction Methods for Rotating Electrical Machines: A Review

Due to the rise of e-mobility applications, there is an increased demand to create more accurate control methods, which can reduce the loss in an e-drive system. The accurate modeling of the rotating machines needs to resolve a partial differential equation system that describes the thermal and mechanical behavior of the different parts in addition to the electromagnetic design. Due to these models’ limited resources and high computation demand, they cannot be used directly for real-time control. Model order reduction methods have been of growing interest in the past decades and offer solutions for this problem. According to the processed literature, many model order reduction-based methods are used for a wide range of problems. However, a paper has not been published that discusses a model order reduction-based real-time control model that is actually used in the industry. This paper aims to summarize and systematically review the model order reduction methods developed for rotating electrical machines in the last two decades and examine the possible usage of these methods for a real-time control problem.

Effect of diffusivity of amyloid beta monomers on the formation of senile plaques.

Alzheimer's disease (AD) presents a perplexing question: why does its development span decades, even though individual amyloid beta (Aβ) deposits (senile plaques) can form rapidly in as little as 24 hours, as recent publications suggest? This study investigated whether the formation of senile plaques can be limited by factors other than polymerization kinetics alone. Instead, their formation may be limited by the diffusion-driven supply of Aβ monomers, along with the rate at which the monomers are produced from amyloid precursor protein (APP) and the rate at which Aβ monomers undergo degradation. A mathematical model incorporating the nucleation and autocatalytic process (via the Finke-Watzky model), as well as Aβ monomer diffusion, was proposed. The obtained system of partial differential equations was solved numerically, and a simplified version was investigated analytically. The computational results predicted that it takes approximately 7 years for Aβ aggregates to reach a neurotoxic concentration of 50 μM. Additionally, a sensitivity analysis was performed to examine how the diffusivity of Aβ monomers and their production rate impact the concentration of Aβ aggregates.

Existence and forms of entire solutions to system of non-linear partial differential equations

Existence and forms of entire solutions to system of non-linear partial differential equations

Forced vibrations of the shaft line based on the model of a two-stage rod with an elastic connection of sections

The paper considers a method for solving the problem of dynamics of forced vibrations of a wall line based on a model of an equivalent elastic two-stage rod with a pliable connection of sections. The model is represented by a system of partial differential equations. The solution of the equations is obtained by the Fourier method for eigenfunctions orthogonal with weight. It is established that the presence of an elastic element at the junction of the sections does not affect the orthogonality of the eigenfunctions. The influence of the elastic coupling compliance coefficient on the natural frequency spectrum of the shaft line is estimated. The obtained forms of natural vibrations of an equivalent elastic two-stage rod are consistent in appearance with the corresponding forms known in the literature for a discrete model of a similar torsional circuit. A partial solution of the equations is found for the case of the action of the harmonic load on the engine and the averaged torque on the propeller. As an example, the dynamics of forced vibrations of the shaft line of a barge of the Sosnovka type is studied. The results of calculating the natural frequencies of the considered model are compared with the classical discrete mass method. The analysis of the obtained plots of the twisting angles and torques showed that the angles are satisfactorily approximated by the first proper oscillation shape, and for moments, high shapes must be taken into account. The comparison of the simulation results with the data of the torsiogram obtained under production testing conditions indicates the need for further development of the model. Further improvement of the torsional circuit, as well as a more detailed study of the nature of the operating loads, are considered as promising areas of research.