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An initial value problem to the viscous Burgers equation under the influence of an external forcing of distribution type is considered. The existence and uniqueness of weak solutions for it are investigated. To do this, we linearize the viscous Burgers equation and show the existence and uniqueness of common boundary function for the associated initial-boundary problems. Jump discontinuity of the spatial derivative of the solution makes a key role in determining large time asymptotics to the solutions. The explicit representation of the common boundary function and its asymptotic behavior are discussed for a specific case of the jump discontinuity. Here, the common boundary is obtained in terms of Associated Legendre functions of the first and second kind. For the general case, asymptotic behavior of the common boundary of the associated initial-boundary problems is used to determine the large time asymptotics for the Cauchy problem to the viscous Burgers equation. [ABSTRACT FROM AUTHOR]

Runge–Kutta (RK) pairs are widely used for numerically solving initial value problems (IVPs), but dealing with step rejections during integration is a common occurrence. Conventionally, when a step is rejected, all calculations made during that step are discarded, and a completely new set of computations is initiated. In our research, we propose a method to address this inefficiency by repurposing the previously computed RK stages from rejected steps. Our primary focus is on the renowned RKF45 pair, consisting of fifth‐ and fourth‐order methods. When a step rejection occurs, we leverage the stages computed in prior steps and introduce just three additional stages. These stages are then used to evaluate the results with a smaller step size. This approach effectively reduces computational costs in various challenging IVPs where RK algorithms with different step sizes encounter difficulties [ABSTRACT FROM AUTHOR]

In this paper, we study a free‐boundary problem modeling the growth of spherically symmetric tumors with angiogenesis and a necrotic core, where the Robin boundary condition is imposed for the nutrient concentration. The existence of a global solution is established by first reducing the free‐boundary problem into an equivalent initial boundary value problem for a nonlinear strongly singular parabolic equation on a fixed domain, then proving that an approximation problem admits a unique solution by the Schauder fixed point theorem combined with the Lp$$ {L}^p $$ estimates for parabolic equations, and finally taking the limit. Compared with the Dirichlet boundary value condition problem, the Robin condition causes some new difficulties in making rigorous analysis of the model, particularly on the uniqueness of solutions to the approximation problem. [ABSTRACT FROM AUTHOR]

This paper establishes the global well‐posedness of strong solutions for the 3D compressible magnetohydrodynamics (MHD) with non‐flat doping profile and Coulomb force, which is of hyperbolic–parabolic–elliptic mixed type. It is essentially shown that for the initial value problem with initial density allowed to contain vacuum states, the strong solution exists globally under small energy but possible large oscillations. [ABSTRACT FROM AUTHOR]

This article presents an efficient one‐step hybrid block method (OHBM) to solve third‐order initial value problems (IVPs). The proposed method is derived using interpolation and collocation techniques of the assumed exact solution and its derivatives. The theoretical properties of the OHBM are analyzed. An embedding‐like technique is utilized and executed in an adaptive mode to improve the performance of the proposed method. The OHBM's efficiency and accuracy are tested by solving practical application problems in applied sciences and engineering. The numerical solutions demonstrate that the proposed OHBM is highly efficient and provides very accurate approximations. [ABSTRACT FROM AUTHOR]

This article studies the initial value problem in an asymptotic autonomous reaction–diffusion system. Namely, when the time goes to infinity, these parameters converge to constants. The system may be nonmonotonic. With different decaying initial conditions, the leftward and rightward spreading speeds are estimated by constructing auxiliary functions and systems. As an application, we present the propagation properties in a diffusive epidemic model with different decaying initial values, which does not satisfy the classical comparison principle of mixed quasimonotone systems. [ABSTRACT FROM AUTHOR]

We reconstruct the initial functions from the trace of the solution of an initial value problem for the wave equation on the light cone. A method to recover the initial function from the solution of the wave equation on the light cone is already known for odd spatial dimensions. We generalize their work to the fractional wave equation and all dimensions. In other words, we present a method to reconstruct the initial functions from the solution of the fractional wave equation on the light cone. [ABSTRACT FROM AUTHOR]

We aim to investigate the stability property for the certain linear and nonlinear fractional q -difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear q -difference equations of the q -Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy q -difference initial value problem, we use the same quantum Laplace transform and the q -Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable. [ABSTRACT FROM AUTHOR]

A system of fractional differential equations (FDEs) with fractional derivatives of diverse orders is called an incommensurate system of FDEs. In this paper, the well-posedness of the initial value problem for incommensurate systems of FDEs is obtained on the space of continuous functions. Three different methods for this analysis are used and compared. The complexity of such analysis is reduced by new techniques. Strong existence results are obtained by weaker conditions. The uniqueness and the continuous dependency of the solution on initial values are investigated using the Gronwall inequality. [ABSTRACT FROM AUTHOR]

In this paper we approach the linear mixed problem with initial and boundary values for a Cosserat body which is elastic and has pores. We coupled the equations characterizing the pores evolution with basic equations that describing the elastic deformations of the Cosserat body. The coupling is done by means of predetermined coefficients. To prove the continuous dependence of the solutions, we introduce an adequate measure which is useful in obtaining of some estimations with respect to the gradient of elastic deformations and regarding the gradient of a specific the application, introduced to characterize the behavior of the pores. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we investigate the flow characteristics of an unsteady quasistatic Maxwell viscoelastic fluid of White-Metzner type within a three-dimensional bounded domain. Our objective is to establish the well-posedness of the resulting partial differential equations initial boundary value problem, both locally and globally in time, under the condition of small initial data. To demonstrate the existence of a solution, we employ an iterative method that transforms our problem into a sequence of linear subproblems. At each iteration, we tackle both elliptic and hyperbolic problems. The convergence of this iterative process provides the solution to the original problem. [ABSTRACT FROM AUTHOR]

We study bifurcations of nonlinear waves (spatially inhomogeneous solutions) emerging from homogeneous equilibrium states of an initial boundary value problem, arising in nonlinear optics, for a nonlinear parabolic equation on a disk with a spatial argument rescaling operator and with time delay. In the plane of the main parameters of the equation, we construct stability (instability) domains of homogeneous equilibrium states and study the dynamics of the stability domains depending on the rescaling coefficient. We investigate the mechanisms of stability loss by homogeneous equilibrium states, the possible bifurcations of spatially inhomogeneous self-oscillatory solutions, and their stability. We demonstrate the possibility of bifurcation of stable rotational and spiral waves. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global spherically symmetric solutions for the initial boundary value problem of a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of two-phase viscous compressible fluids. We prove the existence and uniqueness of global classical solution, weak solution and strong solution under the assumption of spherically symmetry condition for initial data ρ0 without vacuum state. [ABSTRACT FROM AUTHOR]

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation. [ABSTRACT FROM AUTHOR]

Due to the potential for airborne radar to capture incomplete observational information regarding unmanned aerial vehicle (UAV) trajectories, this study introduces a novel approach called Node-former, which integrates neural ordinary differential equations (NODEs) and the Informer framework. The proposed method exhibits high accuracy in trajectory prediction, even in scenarios with prolonged data interruptions. Initially, data outside the acceptable error range are discarded to mitigate the impact of interruptions on prediction accuracy. Subsequently, to address the irregular sampling caused by data elimination, NODEs are utilized to transform computational interpolation into an initial value problem (IPV), thus preserving informative features. Furthermore, this study enhances the Informer's encoder through the utilization of time-series prior knowledge and introduces an ODE solver as the decoder to mitigate fluctuations in the original decoder's output. This approach not only accelerates feature extraction for long sequence data, but also ensures smooth and robust output values. Experimental results demonstrate the superior performance of Node-former in trajectory prediction with interrupted data compared to traditional algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a new method to prove the blowup at $ +\infty $ for the solution of the initial boundary value problem for a class of semilinear parabolic equations with logarithmic nonlinearity at sub-critical initial energy level by contradiction. Moreover, following the idea of contradiction, we obtained a sufficient criterion for the blowup at $ +\infty $ of the solution with arbitrarily positive initial energy by defining a new auxiliary function, in which the location of the initial data concerning the unstable set is not required. [ABSTRACT FROM AUTHOR]

This article is concerned with an appropriate definition of the fractional derivative and integral based on the concept of new generalized Caputo-type fractional derivative introduced recently by Odibat and Baleanu (Appl Numer Math 156:94-105, 2020), hence the abbreviated name OBC, in fuzzy environment. Using this concept, we provide some results on the existence and uniqueness of solutions for a class of fractional fuzzy initial value problems of order m - 1 < κ < m under the newly introduced OBC derivative. Further, we present some applications of the established derivative in the real world with the help of numerical examples based on Euler's method of integration. Along with a theoretical example, we study an example depicting Allee's effect and another model of the growth of technological innovations. [ABSTRACT FROM AUTHOR]

The purpose of this study is to investigate the solutions of initial and boundary value problems of ordinary differential equations by employing Lie symmetry generators. In this investigation, it shown that invariant curves, which obtained by symmetry generators, also be utilized to find solutions to initial and boundary value problems. A method, involving invariant curves, presented to find solutions to initial and boundary value problems. Solutions to many linear and nonlinear initial and boundary value problems discussed by applying the proposed method. [ABSTRACT FROM AUTHOR]

In this manuscript, our work was about a qualitative study for a class of multi-complex orders nonlinear fractional differential equations (FDEs). Our methodology utilized the topological degree theory and studied a novel operator tailored for non-singular FDEs with T-Riemann-Liouville (T-RL) fractional order derivatives. The primary objective was to prove the existence and uniqueness of solutions for both initial and boundary value problems within the intricated framework. With the help of topological degree theory, we contributed to a wider understanding of the structural aspects governing the behavior of the considered FDE. The novel operator proposing for non-singular FDEs added a unique dimension to our analytical problem, offering a versatile and effective means of addressing the challenges posed by these complex systems for their theoretical analysis. For the practical implications of our theoretical framework, we presented two concrete examples that reinforced and elucidated the key concepts introduced. These examples underscored our approach's viability and highlighted its potential applications in diverse scientific and engineering domains. Through this comprehensive exploration, we aimed to contribute significantly to advancing the theoretical foundation related to the study of multi-complex nonlinear FDEs. Moreover, a fixedtime terminal sliding mode control (TSMC) has been developed. This proposed control strategy for eliminating leukemic cells while maintaining normal cells was based on a chemotherapeutic treatment that was not only effective but also widely acknowledged to be safe. This strategy was evaluated using the fixed-time Lyapunov stability theory, and simulations were included to illustrate its performance in terms of tracking and convergence. [ABSTRACT FROM AUTHOR]

In this paper, we are interested to study the initial value problem for parabolic equations with viscoelastic quantity. These types of equations have many applications in physics. Under some appropriate assumptions of the initial datum, we obtain some regularity results of the solution including upper and lower bounds. We achieve some continuous results of the mild solution according to parameter $ k $. Our study also prove that the mild solution (as well as its derivative) of the parabolic memory problem converges to the solution of the classical parabolic problem. [ABSTRACT FROM AUTHOR]

In this paper, the initial value problem of the semilinear σ$$ \sigma $$‐evolution equations with a memory term is concerned. Firstly, using the energy method in the Fourier space, the decay estimates for the solutions to the corresponding linear problem are established. Additionally, assuming small initial data in suitable time‐weighted Sobolev spaces, the global‐in‐time existence of the solutions to the semilinear issue is proved by contraction mapping. Finally, the decay estimates of solutions are obtained under the additional regularity assumption on the initial data. [ABSTRACT FROM AUTHOR]

In this paper, an initial boundary value problem of the 2D incompressible Bénard system is concerned. The global existence of a unique strong solution is established by the energy method and a key logarithmic interpolation inequality. In particular, the exponential decay rates for the gradients of velocity and temperature field are obtained. [ABSTRACT FROM AUTHOR]

A parabolic–elliptic Keller–Segel model for multipopulations in a two‐dimensional unit ball will be considered in this paper. For the initial boundary value problem with mutually attractive populations, the global existence of bounded solutions has been proved through the logarithmic Hardy–Littlewood–Sobolev inequality for system when the initial masses satisfy the subcritical condition. Moreover, based on the moments technique, there exists a blow‐up solution when the initial masses satisfy the supercritical case. In a word, a critical mass phenomenon has been established in this multispecies chemotaxis system. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence and uniqueness of solutions for a system of the first-order q-difference equations with multi-point and q-integral boundary conditions using various fixed point (fp) theorems. Also, we give two examples to support our results. [ABSTRACT FROM AUTHOR]

Ordinary nonlinear differential equations with classical and fractional derivatives are used to simulate several real-world problems. Nonetheless, numerical approaches are used to acquire their solutions. While various have been proposed, they are susceptible to both disadvantages and advantages. In this paper, we propose a more accurate numerical system for solving nonlinear differential equations with classical and Caputo–Fabrizio derivatives by combining two concepts: the parametrized method and the predictor–corrector method. We gave theoretical analyses to demonstrate the method’s correctness, as well as several illustrated examples for both scenarios. [ABSTRACT FROM AUTHOR]

This work explores a well-established eighth-algebraic-order numerical method belonging to the explicit Numerov-type family. To enhance its efficiency, we integrated a cost-effective algorithm for adjusting the step size. After each step, the algorithm either maintains the current step length, halves it, or doubles it. Any off-step points required by this technique are calculated using a local interpolation function. Numerical tests involving diverse problems demonstrate the significant efficiency improvements achieved through this approach. The method is particularly effective for solving differential equations with oscillatory behavior, showcasing its ability to maintain high accuracy with fewer function evaluations. This advancement is crucial for applications requiring precise solutions over long intervals, such as in physics and engineering. Additionally, the paper provides a comprehensive MATLAB-R2018a implementation, facilitating ease of use and further research in the field. By addressing both computational efficiency and accuracy, this study contributes a valuable tool for the numerical analysis community. [ABSTRACT FROM AUTHOR]

We have developed the generalized quasilinearization method (QM) for an initial value problem (IVP) of dynamic equations on time scales by using comparison theorems with a coupled lower solution (LS) and upper solution (US) of the natural type. Under some conditions, we observed that the solutions converged to the unique solution of the problem uniformly and monotonically, and the rate of convergence was investigated. [ABSTRACT FROM AUTHOR]

Analytical solution of initial value problem for the fractional oscillation equation with two Caputo fractional derivatives x′′(t)+aDtαx(t)+cx′(t)+bDtβx(t)+kx(t)=q(t)$$ {x}&#x0005E;{\prime \prime }(t)&#x0002B;a{D}_t&#x0005E;{\alpha }x(t)&#x0002B;c{x}&#x0005E;{\prime }(t)&#x0002B;b{D}_t&#x0005E;{\beta }x(t)&#x0002B; kx(t)&#x0003D;q(t) $$, where the coefficients and orders satisfy a,b,c,k>0,1<α≤2$$ a,b,c,k>0,1<\alpha \le 2 $$ and 0<β≤1$$ 0<\beta \le 1 $$, is investigated by using the Laplace transform and complex inverse integral method on the principal Riemann surface. It is proved by using the argument principle that the characteristic equation has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface under the assumption that α$$ \alpha $$ and β$$ \beta $$ are not both integers. Then three fundamental solutions, the unit impulse response, the unit initial displacement response, and the unit initial rate response, are derived analytically. Each of these solutions is expressed into a superposition of a classical damped oscillation decaying exponentially and a real Laplace integration decaying in a negative power law. Finally, the asymptotic behaviors of these analytical solutions for sufficiently large t$$ t $$ are determined as monotonous decays in a power of negative exponent. [ABSTRACT FROM AUTHOR]

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In this work, we introduce a novel idea of generalized (k , ψ) -Hilfer proportional fractional operators. The proposed operator combines the (k , ψ) -Riemann–Liouville and (k , ψ) -Caputo proportional fractional operators. Some properties and auxiliary results of the proposed operators are investigated. The ψ-Laplace transform and its properties of the proposed operators are established and utilized to solve Cauchy-type problems. Furthermore, the uniqueness result for a higher-order initial value problem under (k , ψ) -Hilfer proportional fractional operators is proved by using Picard's iterative technique. At the end, examples are provided to present the theoretical results. This new type of proposed operator can help other researchers who are still working on real-world problems. [ABSTRACT FROM AUTHOR]

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global existence of large solutions to the initial and initial boundary value problems for the radiation hydrodynamics model with constant viscosity coefficient and temperature-dependent heat conductivity coefficient. In particular, the initial data and γ - 1 could be arbitrarily large. Compared with the classical compressible Navier--Stokes equations, the research on the radiation hydrodynamics model is more complicated due to the presence of the radiation effect. To overcome difficulties caused by the radiation, inspired by [J. Li and Z. L. Liang, Arch. Ration. Mech. Anal., 220 (2016), pp. 1195--1208; J. H. Zhang and H. J. Zhao, SIAM J. Math. Anal., 55 (2023), pp. 6229--6261], we construct a pointwise estimate between the radiative heat flux and the temperature; then we can establish the desired basic energy estimate by considering that the temperature has lower and upper bounds separately. And once that is obtained, the main result is proved by employing the elementary energy methods. [ABSTRACT FROM AUTHOR]

We shall prove the global existence of weak solutions to an initial boundary value problem for a novel phase-field model which is an elliptic-parabolic coupled system. This model is proposed as an attempt to describe the motion of grain boundaries, a type of interface motion by interface diffusion driven by bulk free energy in elastically deformable solids. Its applications include important processes arising in materials science, e.g., sintering. In this model the evolution equation for an order parameter is a nonuniformly, degenerate parabolic equation of fourth order, which differs from the Cahn--Hilliard equation by a nonsmooth term of the gradient of the unknown. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to proving some results concerning the existence of solutions for a class of initial value problem for nonlinear implicit fractional Hybrid differential equations and improved conformable fractional derivative. The result is based on a fixed point theorem due to Dhage. Further, an example is provided for the justification of our main result. [ABSTRACT FROM AUTHOR]

Recently, a stability theory has been developed to study the linear stability of modified Patankar–Runge–Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the convergence of an MPRK scheme towards the steady state of the corresponding initial value problem, whereas the main assumption is that the initial value is sufficiently close to the steady state. Initially, numerical experiments in several publications indicated that these linear stability properties are not only local but even global, as is the case for general linear methods. Recently, however, it was discovered that the linear stability of the MPDeC(8) scheme is indeed only local in nature. Our conjecture is that this is a result of negative Runge–Kutta (RK) parameters of MPDeC(8) and that linear stability is indeed global if the RK parameters are nonnegative. To support this conjecture, we examine the family of MPRK22( α ) methods with negative RK parameters and show that even among these methods there are methods for which the stability properties are only local. However, this local linear stability is not observed for MPRK22(α ) schemes with nonnegative Runge–Kutta parameters. In particular, it is shown that MPRK22(α ) schemes with 0 < α < 0.5 or - 0.5 < α < 0 are only stable if the time step size is sufficiently small. But schemes with α ≤ - 0.5 are stable and converge towards the steady state of the initial value problems for all time step sizes, at least for the test problem under consideration. Furthermore, it is shown that for some of the latter systems, the initial values must actually be close enough to the steady state to guarantee this result. [ABSTRACT FROM AUTHOR]

In this study, we significantly extend the concept of modular metric-like spaces to introduce the notion of b-metric-like spaces. Furthermore, by incorporating a binary relation R , we develop the framework of R -modular b-metric-like spaces. We establish a groundbreaking fixed point theorem for certain extensions of Geraghty-type contraction mappings, incorporating both simulation function and E -type contraction within this innovative structure. Moreover, we present several novel outcomes that stem from our newly defined notations. Afterwards, we introduce an unprecedented concept, the graphical modular b-metric-like space, which is derived from the binary relation R. Finally, we examine the existence of solutions for a class of functional equations that are pivotal in dynamic programming and in solving initial value problems related to the electric current in an RLC parallel circuit. [ABSTRACT FROM AUTHOR]

In this study, we derive an analytical solution to address the problem of vertical infiltration within 1D homogeneous bounded profiles. Initially, we consider the Richards equation together with Dirichlet boundary conditions. We assume constant diffusivity and linear dependence between the conductivity and the water content, resulting to a linear partial differential equation of diffusion type. To solve the simplified initial boundary value problem over a finite interval, we apply the unified transform, commonly known as the Fokas method. Through this methodology, we obtain an integral representation of the solution that can be efficiently and directly computed numerically, yielding a convergent scheme. We examine various cases, and we compare our solution with well‐known approximate solutions. This work can be seen as a first step to derive analytical solutions for the far more difficult and complex problem of modelling water flow in heterogeneous layered soils. [ABSTRACT FROM AUTHOR]

We introduce a regularization method with two differential operators for solving a linear ill-posed operator equation system. The existence and uniqueness of regularized solutions to the problem are derived. With an a priori as well as an a posteriori parameter choice strategy, convergence and convergence rates of the regularized solution are also obtained. As an application, we apply the regularization to a simultaneous inversion of the source term and the initial value problem for a heat conduction equation, numerical experiments are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The paper is concerned with the study of a chemical reaction between diffusing and solid constituents. The reaction is localized at a propagating reaction front and is accompanied by a transformation strain, which generates stresses affecting the reaction rate. The effect of mechanical stresses on the velocity of the reaction front is taken into account basing on the concept of a chemical affinity tensor. The initial boundary value problem of the propagation of the chemical reaction front is discussed by the example of an axisymmetric problem for a reaction in a cylinder in the case of linear elastic solid constituents. The stage of initial accumulation of the diffusing constituent and the stage of the propagation of the reaction front after its separation from the outer boundary of a body are distinguished. The time preceding the start of the reaction is determined. Conditions at the propagating chemical reaction front for the diffusion problem are discussed. A comparative analysis of the solutions obtained in the quasi-stationary and nonstationary formulations of the diffusion problem is carried out, taking into account and without taking into account the influence of the initially accumulated diffusing constituent on the reaction front velocity, as well as for the case of the absence of diffusion in the untransformed material. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel series of second-order Backward Differentiation Formulas (BDFs) specifically designed to address phase-lag and its first derivative in the numerical resolution of Initial Value Problems (IVPs) with orbital solutions. Our methodology commences with an in-depth analysis of phase-lag phenomena associated with second-order BDFs. Following this, we construct a suite of equations by embedding algebraic functions into the operational framework of the 3-step second-order BDF (SOBDF) method. Additionally, we elaborate on equations that precisely describe the phase-lag and its derivatives, with a concentrated focus on the 3-step SOBDF method. The culmination of this work is the presentation of six distinct methods, each methodically crafted to negate both the real and imaginary elements of phase-lag and its derivatives in numerical computations. The study advances with a meticulous examination of the local truncation error and the stability regions pertinent to the six phase-fitted methods introduced. Furthermore, we scrutinize their computational performance by deploying these methods across a spectrum of initial value problems, offering valuable insights into their effectiveness in varying contexts. [ABSTRACT FROM AUTHOR]

This work aims to implement a one-step block method with three optimized hybrid points for solving singularly perturbed nonlinear initial value problems on non-uniform meshes. The basic properties of the method are investigated and it is shown that the proposed method is consistent, zero-stable, convergent and has a large region of absolute stability. The presented numerical results confirm the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider large-scale low-rank Sylvester differential matrix equations. We present two iterative methods for the approximate solution of such difierential matrix equations. The first approach is based on the integral expression of the exact solution which exploits an extended block Krylov subspace method to compute the exponential of a matrix times a block of vectors. In the second method, we first project the initial value problem onto an extended block Krylov subspace and acquire a low-dimensional Sylvester difierential matrix equation with a low-rank constant term. Then the reduced Sylvester difierential matrix equation is solved by the backward difierentiation formula method (BDF) and the derived solution is used to construct the low-rank approximate solutions of the original initial value problem. The iterative approaches are followed until some certain accuracy is obtained. We give some theoretical results and some numerical examples to show the eficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

We are concerned with the long time existence of strong solutions to the initial boundary value problem with slip boundary conditions for the compressible Navier-Stokes equations in three-dimensional simply connected bounded domains. We verify that the barotropic Navier-Stokes equations admit a unique strong solution on the time interval where the solution to the incompressible Navier-Stokes equations exists and is reasonably smooth, when the Mach number is sufficiently small. Furthermore, we obtain the uniform convergence of strong solutions for the compressible system toward those for the corresponding incompressible system on that time interval. [ABSTRACT FROM AUTHOR]

This paper provides the process of incremental constitutive integration for the unified hardening model combined with the transformation stress method. The dimension-extending technique takes the hardening function of the hardening/softening model as the same position as the stress components, so that the constitutive integration of the plasticity can be reduced to an initial value problem of differential–complementarity equations, which is solved using the Gauss–Seidel algorithm-based Projection–Correction for the mixed complementarity problem. The Gauss–Seidel based Projection–Correction algorithm does not require the calculation of the Jacobean matrix of the potential function, making it relatively easy to implement in programming. The unified hardening model is proposed based on the modified Cam–Clay model and the sub-loading surface model, and the elastic properties are pressure-dependent. Two processing methods, backward Euler integration and exact elastic property, are used for the variable elasticity properties. The constitutive integration of the increased dimensional unified hardening model is reduced to a special mixed complementarity problem and solved by the proposed algorithm, which does not need to calculate the Jacobean matrix of the potential function, and greatly simplifies the derivation process. Several numerical examples are given to verify the feasibility of the incremental constitutive integration in the unified hardening model, including the single integral point and the boundary value problems. The research results have expanded the scope of use of the Gauss–Seidel based Projection–Correction algorithm. [ABSTRACT FROM AUTHOR]

The article studies quantitative relations (error identities) that characterize distances between exact solutions of nonlinear evolutionary problems and functions considered as approximations. The restrictions imposed on such a function are minimal and actually come down to the condition that it belongs to the same functional class as the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as for analysis of modeling errors. The corresponding examples are discussed in the paper. [ABSTRACT FROM AUTHOR]

We consider the initial value problem for an ordinary differential equation with distributed order differentiation operator regarded as the integral of the Riemann–Liouville fractional derivative with respect to the order of differentiation over a Borel measure with compact support. We construct the fundamental solution to the equation in terms of the Wright function with distributed parameters and obtain the general representation for solutions to this equation. We prove the existence and uniqueness theorem for the Cauchy problem and express the solution in an explicit form. [ABSTRACT FROM AUTHOR]

Application: This study focuses on the analysis of nanofluids, which are suspensions of nanoparticles in standard fluid transfer systems. The investigation specifically examines the behavior of nanoparticles over a deformable surface. Purpose and Methodology: Thermal radiation and magnetohydrodynamic forms are incorporated into the energy and momentum equations, respectively. Basic fluid mechanics principles are employed to derive the governing flow equations. The use of similarity transforms facilitates the transformation of ordinary differential equations into partial differential equations. Nonlinear ordinary differential equations are then reduced to a set of initial value problems using the shooting technique. Core Findings: Numerical results are obtained for the thermal profile. The study indicates thermal optimization through the combined inclusion of water with alumina and polymers with copper oxide and titanium dioxide nanoparticles. Graphical analysis explores the absorption of parameters in velocity and temperature profiles. Future Work: Future research could delve deeper into the optimization of thermal properties and investigate additional nanoparticle combinations. Further exploration of the effects of different surface deformations and variations in fluid properties could also be pursued. [ABSTRACT FROM AUTHOR]

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions. [ABSTRACT FROM AUTHOR]

We provide here a novel approach for solving IVPs in ODEs and MTFDEs numerically by means of a class of MSJPs. Using the SCM, we build OMs for RIs and RLFI for MSJPs as part of our process. These architectures guarantee accurate and efficient numerical computations. We provide theoretical assurances for the efficacy of an algorithm by establishing its convergence and error analysis features. We offer five numerical examples to prove that our method is accurate and applicable. Through these examples, we demonstrate the greater accuracy and efficiency of our approach by comparing our results with previously published findings. Tables and graphs show that the method produces exact and approximate solutions that agree quite well with each other. [ABSTRACT FROM AUTHOR]