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This article reports on a qualitative investigation into students' thinking about a differential equations problem posing task; i.e. an initial value problem. Analysis of written and verbal responses to the task indicate that only four of the 34 students who participated in the study were successful in posing problems. Furthermore, only one of the problems they posed is solvable. In general, findings of this study indicate that posing problems involving initial value problems is particularly challenging for students. In light of the multitude of benefits of problem posing in mathematics education, I recommend the inclusion of problem posing tasks in the teaching and learning of undergraduate mathematics courses, including ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial and boundary value problem to the three-dimensional (3D) nonhomogeneous nematic liquid crystal flows with density-dependent viscosity and vacuum. Combining delicate energy method with the structure of the system under consideration, the global well-posedness of strong solutions is established, provided that ‖ ρ 0 ‖ L 1 + ‖ ∇ d 0 ‖ L 2 is suitably small. In particular, the initial velocity can be arbitrarily large. Moreover, the exponential decay rates of the strong solution are also obtained. [ABSTRACT FROM AUTHOR]

This paper deals with the following indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source u t = Δ u v α + a u - b u l , x ∈ Ω , t > 0 , v t = Δ v - v w , x ∈ Ω , t > 0 , w t = - δ w + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n ( n ≥ 1 ). Here, the parameters a > 0 , b > 0 , α ≥ 1 , δ > 0 and l ≥ 2 . For all suitably regular initial data, if one of the following cases holds: l > 2 ; l = 2 , n ≤ 3 ; l = 2 , n ≥ 4 , and b is sufficiently large, then the corresponding initial boundary value problem possesses a global classical solution. [ABSTRACT FROM AUTHOR]

We present two new methods, a simple fast and a slower more precise one, to reduce the truncation error occurring during multiplication in verified Taylor model arithmetic. These methods were implemented in MATLAB in INTLAB's Taylor model toolbox which targets solving ordinary differential equations rigorously, i.e., numerical solutions are computed along with rigorous error bounds that include all numerical as well as all rounding errors so that the exact solution must lie within these error bounds. The methods are applied to several test cases to show their effect. [ABSTRACT FROM AUTHOR]

Applying a method with vanished phase–lag might potentially eliminate the phase–lag and its first, second, and third derivatives. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost–efficient approach. Equation PF3DPHFITN142SPS demonstrates the unique method. The suggested approach is P–Stable, meaning it is indefinitely periodic. The suggested approach is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost–efficient approach. With an AOR of 14, we can significantly improve our present predicament. [ABSTRACT FROM AUTHOR]

Solutions of ordinary differential equations are considered. Differential Space is defined as the three-dimensional space with coordinates being the solution function and its first and second derivatives. Solution curves are represented as parametric three-dimensional curves in the differential space with the curve parameter being the independent variable. For various sample differential equations, the solution curves and their properties are depicted. The solution curves may converge to a point, may blow up and diverge to infinity, may be periodic, may end up with a limit cycle periodic solution or may be chaotic. Local solutions with a given initial condition set is treated in this introductory study. Differential space is a generalization of 2-D state space to 3-D’s. Global solutions with phase diagrams, basins of attraction are left for further studies. [ABSTRACT FROM AUTHOR]

We concern the structural stability of supersonic flows with a contact discontinuity in a finitely long curved nozzle for the two‐dimensional steady compressible rotating Euler system. Concerning the effect of Coriolis force, we first establish the existence of supersonic shear flows with a contact discontinuity in the flat nozzle. We then investigate the structural stability of these background supersonic shear flows with a contact discontinuity under the perturbations of the incoming supersonic flow and the upper and lower nozzle walls. It can be formulated as an initial boundary value problem with a contact discontinuity as a free boundary. The Lagrangian transformation is employed to straighten and fix the contact discontinuity, and the rotating Euler system is reduced to a first‐order hyperbolic system for the Riemann invariants. The key ingredient of the analysis is to obtain the estimates of the solution for the associated linearized hyperbolic system via the characteristic method. [ABSTRACT FROM AUTHOR]

This study presents a novel exploration into the intricate interplay of slanted magnetohydrodynamic and radiative fluxes within a Williamson fluid incorporating single‐walled carbon nanotubes (SWCNTs) and multi‐walled carbon nanotubes (MWCNTs) in a water‐based system. Utilizing an exponentially stretching cylinder model, the research holistically integrates entropy production, encompassing thermal radiation, porosity, buoyant forces, and heat source/sink constraints within the governing equations. To solve the resulting governing equations with boundary conditions, the shooting technique is employed to convert them into initial value problems. Subsequently, the Runge–Kutta–Fehlberg 4–5th order method is used to obtain numerical solutions. Unveiling unprecedented insights, the investigation reveals distinctive parameter dependencies governing fluid behavior. It delineates how variations in different factors distinctly impact fluid velocity, thermal profiles, entropy generation, Bejan number, drag force, and heat transfer rates across different nanofluid compositions. Furthermore, the study highlights the transformative impact of nanocomposites on altering the thermal properties of the fluid and orchestrating nanofluid mobility. The results show that MWCNTs have an enhanced entropy generation rate when compared to SWCNTs. These findings bear substantial promise for real‐world applications involving water‐based nanofluids, shedding light on their manipulation and optimization in various technological domains. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to develop a new class of sixth- and fourth-order implicit Runge–Kutta methods to treat linear and nonlinear initial value ordinary differential equations (ODEs) in applied science and engineering. In this regard, a completely unconventional approach is taken to derive the new formulations. They are derived through a computational approach without involving in the rigorous theory of order conditions. The new formulation has two distinctive parts, namely integration and interpolation. In the first part, we introduce a form of multi-stage single-step numerical integrator with unknown weights. These weights are determined by four simple interconnectivity rules that relate them to minimize the error of the integrator. Some of these rules are based on our expertise gathered from the available prominent quadratures. However, the most important one, the so-called fundamental weighting rule (FWR), is obtained from evolutionary and statistical computations. The second part of this research focuses on developing efficient tools for evaluating the internal stages of the integrator. Accordingly, a series of strong Hermite interpolators has been extended to approximate the solution values at the internal stages. They effectively collaborate with the integrator in several numerical algorithms, the so-called α I I - (q + r) P algorithms, to solve applied ODEs. Development of the integrator, its weighting rules, and the interpolators constitutes the central contribution of this research. Numerical studies demonstrate that the presented algorithms exhibit a high level of accuracy when approximating high-frequency problems as well as long-term solutions of linear and nonlinear ODEs. [ABSTRACT FROM AUTHOR]

In this paper, we consider a pseudo‐parabolic equation with the Caputo fractional derivative. We study the existence and uniqueness of a class of mild solutions of these equations. For a nonlinear problem, we first investigate the global solution under the initial data u0 ∈ L2. In the case of initial data u0 ∈ Lq, q ≠ 2, we obtain the local existence result. Our main tool here is using fundamental tools, namely, Banach fixed point theorem and Sobolev embeddings. In addition, we give an example to illustrate the effectiveness of the method has been proposed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Bresse‐Timoshenko‐type system with distributed delay term. Under suitable assumptions, we establish the global well‐posedness of the initial and boundary value problem by using the Faedo‐Galerkin approximations and some energy estimates. By using the energy method, we show two exponential stability results for the system with delay in vertical displacement and in angular rotation, respectively. This extends earlier results in the literature. [ABSTRACT FROM AUTHOR]

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known. [ABSTRACT FROM AUTHOR]

We develop a Riemann-Hilbert approach to the modified focusing complex short pulse (mfcSP) equation Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. with zero boundary conditions (as |x| → ∞) We obtain a parametric representation of the solution of the initial value problem for the mfcSP equation in terms of the solution of the associated Riemann-Hilbert problem. This representation is then used for retrieving one-soliton solutions. [ABSTRACT FROM AUTHOR]

We present a new family of ninth‐order hybrid explicit Numerov‐type methods, effectively utilizing only eight stages, for solving the special second‐order initial value problem. After applying a number of simplifying assumptions, we arrive to a reduced set of order conditions. Then, we derive an optimal method with constant coefficients that requires one less stage than standard methods found in the literature that use nine stages at this moment. Numerical tests are conducted using quadruple precision arithmetic on several well‐known problems and the superiority of the new method is clear. Finally, in Section 6, a Mathematica package is presented that implements the corresponding algorithm. [ABSTRACT FROM AUTHOR]

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation. [ABSTRACT FROM AUTHOR]

The inverse scattering transform is a tool for solving the initial value problem of nonlinear equations, and the solution of the initial-value problem by inverse scattering transform proceeds in three steps: direct scattering, time evolution and inverse problem. In this paper, we discuss the higher-order integrable discrete nonlinear Schrödinger equation which is subjected to inverse scattering transform under nonzero boundary conditions, and the corresponding soliton solutions are obtained and illustrated. [ABSTRACT FROM AUTHOR]

In the present paper we investigate three-dimensional dynamical mixed boundary-transmission problems for composed body consisting of two adjacent anisotropic elastic components having a common interface surface. The two contacting elastic components are subject to different mathematical models: Green–Lindsay’s model of generalized thermo-electro-magneto-elasticity in one component and Green–Lindsay’s model of generalized thermo-elasticity in the other one. We assume that the composed elastic structure under consideration contains an interfacial crack. We prove the uniqueness and existence theorems in appropriate function spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the initial value problem of Caputo tempered fractional stochastic differential equations and then study the well-posedness of its solution. Further, a Euler–Maruyama (EM) method is derived for solving the considered problem. The strong convergence order of the derived EM method is proved to be $ \alpha -\frac {1}{2} $ α − 1 2 with $ \frac {1}{2} \lt \alpha \lt 1 $ 1 2 < α < 1. Additionally, a fast EM method is also developed which is based on the sum-of-exponentials approximation. Finally, numerical experiments are given to support the theoretical findings of the above two methods and verify computational efficiency of the fast EM method. The fast EM method can greatly improve the computational performance of the original EM method. [ABSTRACT FROM AUTHOR]

Due to their several attractive properties, BDF-type multistep methods are usually the method-of-choice for solving stiff initial value problems (IVPs) of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs). Recently, a class of BDF-type methods based on linear barycentric rational interpolants (LBRIs), referred to as RBDF methods, was introduced for solving ODEs. In the present paper, we are going to introduce a new family of LBRIs-based BDF-type formulas for the numerical solution of ODEs and DAEs with desirable stability properties and smaller error constants than those of the RBDF methods. Numerical experiments of the proposed methods on some well-known IVPs for ODEs and DAEs with index ≤ 3 illustrate the efficiency and capability of the methods in solving such problems. [ABSTRACT FROM AUTHOR]

Applying a phase-fitting method might potentially vanish the phase-lag and its first derivative. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost-efficient approach. Equation PF1DPHFITN142SPS demonstrates the unique method. The suggested approach is P-Stable, meaning it is indefinitely periodic. The proposed method is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost-efficient approach. With an AOR of 14, we can significantly improve our present predicament. [ABSTRACT FROM AUTHOR]

Applying a phase-fitting method might potentially vanish the phase-lag and its first derivative. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost-efficient approach. Equation PF2DPHFITN142SPS demonstrates the unique method. The suggested approach is P-Stable, meaning it is indefinitely periodic. The proposed method is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost-efficient approach. With an AOR of 14, we can significantly improve our present predicament. [ABSTRACT FROM AUTHOR]

This paper presents the dynamics that one microorganism spatially expands with two nutrients in space-time periodic environment. The generalized principal eigenvalue defined by the linear periodic parabolic operator is applied as a threshold to discuss the associated initial value problem. When the generalized principal eigenvalue is nonnegative, the solutions approach to the microorganism-extinction equilibrium uniformly. When the generalized principal eigenvalue is negative, the spreading speeds of the model are established when the initial distribution of microorganism has nonempty compact support, which are determined by a family of periodic parabolic eigenvalue problems. In the homogeneous environment, we show that the solutions locally uniformly converge to the microorganism-existent steady state by constructing upper and lower solutions. Finally, numerical simulations are performed to illustrate that the leftward or rightward spreading speed is approximately a constant under various circumstances. [ABSTRACT FROM AUTHOR]

In this paper, the notion of C-semigroup of continuous module homomorphisms on a complete random normal (RN) module is introduced and investigated. The existence and uniqueness of solution to the Cauchy problem with respect to exponentially bounded C-semigroups of continuous module homomorphisms in a complete RN module are established. [ABSTRACT FROM AUTHOR]

In this paper a hyperbolic system of partial differential equations for two-phase mixture flows with N components is studied. It is derived from a more complicated model involving diffusion and exchange terms. Important features of the model are the assumption of isothermal flow, the use of a phase field function to distinguish the phases and a mixture equation of state involving the phase field function as well as an affine relation between partial densities and partial pressures in the liquid phase. This complicates the analysis. A complete solution of the Riemann initial value problem is given. Some interesting examples are suggested as benchmarks for numerical schemes. [ABSTRACT FROM AUTHOR]

The aim of the present study is to investigate the damping of slow sausage MHD waves propagating in a gravitationally-stratified magnetic cylindrical structure when the plasma is strongly partially ionised. The problem is treated as an initial value problem and the analysis deals with the temporal evolution of waves in an asymptotic sense, i.e., large values of time compared to the period of waves. The plasma is assumed to be collision-dominated, i.e., we employ a two-fluid approximation. The set of equations describing the plasma dynamics is reduced to a coupled partial differential equations. Our findings show that the slow wave of charged species is affected by the presence of a cut-off. The mode associated with the neutral fluid propagates without any cut-off and decay very quickly due to collisions between particles. [ABSTRACT FROM AUTHOR]

In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity. [ABSTRACT FROM AUTHOR]

Second order Ordinary Differential Equations (ODE) were considered. Numerov-like techniques employing effectively seven stages per step and sharing eighth algebraic order were under examination for numerically solving them. The coefficients of these methods were contingent on four independent parameters. To tackle issues with oscillatory solutions, we typically aimed to fulfill specific criteria such as minimizing phase-lag, expanding the periodicity interval, or even neutralizing amplification errors. These latter attributes stemmed from a test problem mimicking an ideal trigonometric trajectory. Here, we suggested training the coefficients of the chosen method family across a broad spectrum of pertinent problems. Following this training using the differential evolution method, we identified a particular method that surpassed others in this category across an even broader array of oscillatory problems. [ABSTRACT FROM AUTHOR]

This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model’s parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term. [ABSTRACT FROM AUTHOR]

In this article, we first convert a second order singularly perturbed boundary value problem (SPBVP) into a pair of initial value problems, which are solved later using exponential time differencing (ETD) Runge–Kutta methods. The stability analysis of the proposed scheme is addressed. Some linear and non-linear problems have been solved to study the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Recently, one artificial intelligence technique, known as artificial neural network (ANN), has brought advanced development to the arena of mathematical research. It competes effectively with other traditional methods in providing accurate solutions for fractional differential equations (FDEs). This work aims to implement a feedforward ANN with two hidden layers to solve nonlinear systems based on the fractional Riccati differential equation (FRDE). The network parameters are trained using the Adam optimization method with the aid of automatic differentiation. A vectorization algorithm is designated for the selected step to make the computation process more efficient. Two different initial value problems in integer-order derivatives and fractional-order derivatives are discussed. Numerical results demonstrate that the proposed method not only closely matches the exact solutions and reference solutions but also is more accurate than other existing methods. [ABSTRACT FROM AUTHOR]

It is effective to consider discretized problems for partial differential equations that are difficult to analyze themselves. The global solvability of the initial value problem of the Kirchhoff equation is still unsolved except in cases where the initial value is a specially restricted class. In this paper, we prove the global solvability of the semidiscrete Kirchhoff equation, which is obtained by discretizing the Kirchhoff equation with respect to spatial variables. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse problem of determining the initial value of the distributed‐order time‐fractional diffusion equation from the final time observation data, which arises in some ultra‐slow diffusion phenomena in applied areas. Since the problem is ill‐posed, we propose an iterated regularization method based on the Nesterov acceleration strategy to deal with it. Convergence rates for the regularized approximation solution are given under both the a priori and a posteriori regularization parameter choice rules. It is shown that the proposed method can always yield the order optimal convergence rates as long as the iteration parameter which appears in the Nesterov acceleration strategy is chosen large enough. In numerical aspect, the main advantage of the proposed method lies in its simplicity. Specifically, due to the Nesterov acceleration strategy, only a few number of iteration steps are required to obtain the approximation solution, and at each iteration step, we only need to numerically solve the standard initial‐boundary value problem for the distributed‐order time‐fractional diffusion equation. Some numerical examples including one‐dimensional and two‐dimensional cases are presented to illustrate the validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The paper is concerned with error identities for a class of parabolic equations. One side of such an identity is a natural measure of the distance between a function in the corresponding energy class and the exact solution of the problem in question. Another side is either directly computable or serves as a source of fully computable error bounds. Particular forms of the identities can be viewed as analogs of the hypercircle identity well known for elliptic problems. It is shown that identities possess an important consistency property. Therefore, the identities and the corresponding error estimates can be used in quantitative analysis of direct and inverse problems associated with parabolic equations. The first part of the paper deals with linear parabolic equations. A class of nonlinear problems is considered in the second part. In particular, this class includes problems whose spatial parts are presented by the α-Laplacian operator. [ABSTRACT FROM AUTHOR]

Summary In this paper, computational analysis of a Caputo fractional variable‐order system with inductor‐capacitor (LC) and resistor‐capacitor (RC) electrical circuit models is presented. The existence and uniqueness of solutions to the given problem are determined using Schaefer's fixed point theorem and the Banach contraction principle, respectively. The proposed problem's computational consequences are addressed and analyzed using modified Euler and Runge–Kutta fourth‐order techniques. Furthermore, the suggested model compares several orders, including integer, fractional, and variable orders. To demonstrate the utility of the proposed approach, computational simulations are carried out on LC and RC circuit models of various orders. Furthermore, a comparative analysis with previous investigations has been carried. For the given problem, the numerical solution results in high‐precision approximations. [ABSTRACT FROM AUTHOR]

We demonstrate that a convex combination of the Nagumo and Perron growth conditions guarantees the existence of a unique solution to an ordinary differential equation, which can be obtained as the uniform limit of a sequence of successive approximations. Furthermore, we give a sufficient condition for the pathwise uniqueness of solutions to stochastic differential equations driven by Brownian motions. [ABSTRACT FROM AUTHOR]

In the present work, the transmutation operator approach is employed to construct an exact solution to the initial boundary-value problem for multidimensional free transverse equation vibration of a thin elastic plate with a singular Bessel operator acting on geometric variables. We emphasize that multidimensional Erdélyi–Kober operators of a fractional order have the property of a transmutation operator, allowing one to transform more complex multidimensional partial differential equations with singular coefficients acting over all variables into simpler ones. If th formulas for solutions are known for a simple equation, then we also obtain representations for solutions to the first complex partial differential equation with singular coefficients. In particular, it is successfully applied to the singular differential equations, particularly when they involve operators of the Bessel type. Applying this operator simplifies the problem at hand to a comparable problem, even in the absence of the Bessel operator. An exact solution to the original problem is constructed and analyzed based on the solution to the supplementary problem. [ABSTRACT FROM AUTHOR]

This paper proposes a novel yet simple approach to the adaptive finite element (FE) analysis of the first-order Initial Value Problems (IVPs) in the maximum norm by introducing the reduced element technique. In the present approach, the FE solution uh of the conventional Galerkin element of degree m + 1 is decomposed into two parts: a reduced solution u R h from the reduced element of degree m obtained by ignoring the highest degree term of uh, and a built-in point-wise error estimator ε R h directly given by the ignored term. Since the end node solutions of the reduced element are inherited from the full order element, it gains O(h2m+2) accuracy and achieves a nodal/element accuracy ratio as high as two, which greatly enhances its adaptive capability regarding solving IVPs on long time domains. The related error analysis is addressed and a complete adaptivity algorithm is given. Typical numerical examples of both linear and nonlinear IVPs of both single and systems of equations are presented to show the validity and effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Lyapunov type integral inequality is one of the important inequalities in qualitative analysis of differential equations. In this paper, using properties of conformable derivatives and integral, and the solution of initial value problem for conformable fractional differential equation, some new Lyapunov type integral inequalities for higher order nonlinear fractional Hamiltonian systems are obtained, which improve and unify some known results in the literatures. [ABSTRACT FROM AUTHOR]

The phase-lag and all of its derivatives (first, second, third, fourth, fifth, and sixth) might be eliminated using a phase-fitting technique. The new approach, which is referred to as the economical method, targets maximizing algebraic order (AOR) and reducing function evaluations (FEvs). The one-of-a-kind approach is demonstrated by Equation PF6DPFN142SPS.The proposed method is infinitely periodic i.e. P-Stable. To many periodic and/or oscillatory problems, the suggested strategy can be applied. Using this innovative method, the difficult issue of Schrödinger-type coupled differential equations was tackled in quantum chemistry. Every step of the new approach only requires 5FEvs to execute, making it a economic algorithm. By accomplishing a AOR of 14, this allows us to greatly enhance our current situation. [ABSTRACT FROM AUTHOR]

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker L p -Carathéodory condition. Existence of solutions on an interval [0, T] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities. [ABSTRACT FROM AUTHOR]

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial "volume" problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities. [ABSTRACT FROM AUTHOR]

Recently, the author developed a theory for the computation of the phase lag and amplification factor for explicit and implicit multistep methods for first-order differential equations. In this paper, we will investigate the role of the derivatives of the phase lag and the derivatives of the amplification factor on the efficiency of the newly developed methods. We will also present the stability regions of the newly developed methods. We will also present numerical experiments and conclusions on the newly developed methodologies. [ABSTRACT FROM AUTHOR]

In 2015, Abdeljawad defined the conformable fractional derivative (Grunwald–Letnikov technique) to iterate the conformable fractional integral of order 0 < α ≤ 1 (Riemann approach), yielding Hadamard fractional integrals when α = 0 . The Gronwall type inequality for generalized operators unifying Riemann–Liouville and Hadamard fractional operators is obtained in this study. We use this inequality to show how the order and initial conditions affect the solution of differential equations with generalized fractional derivatives. More features for generalized fractional operators are established, as well as solutions to initial value problems in several new weighted spaces of functions. [ABSTRACT FROM AUTHOR]

Contributing to research on undergraduate students' thinking about problem solving tasks, the present study reports on students' reasoning about two initial-value problems i.e. first-order linear ordinary differential equations with initial conditions. A qualitative analysis of task-based interviews and work written by 34 students revealed that although the students knew the algebraic steps they had to use to solve the two problems, 16 students reported that steps involving integral calculus were particularly challenging. In fact, a majority of these students exhibited lack of facility with the $ u $ u -substution integration technique, something that severed as a stumbling block for them when calculating the integrating factor in their attempt to solve the two problems. Only seven students were successful in mathematizing one of the problems. Five students created and used diagrams to support their thinking when solving one of the problems. These students reported that the creation of the diagrams helped them to solve one of the two problems. Overall, findings of this study suggest that helping students develop greater facility with integration techniques, and more broadly integral calculus, could increase students' success when solving initial-value problems. Implications for instruction and directions for future research are discussed. [ABSTRACT FROM AUTHOR]

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$$ {L}&#x0005E;p $$ 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]