Initial Value Problem Research Articles

Large-time behavior of the solutions for an inflow problem of the full compressible Navier–Stokes–Korteweg equations

In this article, we are concerned with the large-time behavior of solutions to an inflow problem for the one-dimensional full compressible Navier–Stokes–Korteweg equations, which can be used to model the compressible viscous fluids with internal capillarity and heat conductivity. Under some suitable assumptions of the far fields and boundary values of the density, the velocity and the absolute temperature, the large-time behavior of the solutions for the initial boundary value problem is govern by the stationary solution and basic waves of the corresponding hyperbolic system. Then the asymptotic stability of the boundary layer, and the superposition of the boundary layer and the three-rarefaction wave are shown provided that the initial perturbation and the strength of the rarefaction wave and stationary wave are small. The proof is mainly based on L2-energy method, some time-decay estimates in Lp-norm for the smoothed rarefaction wave and the space variable decay estimates of the boundary layer. This can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the inflow problem of the full compressible Navier–Stokes–Korteweg equations.

Dynamical response of a floating ice sheet due to a forced oscillation in a running stream

This paper explores the response of a floating ice sheet to a forced, time-harmonic oscillatory pressure applied to the ice-covered surface of a uniform, finite depth fluid. The floating ice sheet is modeled as a thin elastic plate following the Euler–Bernoulli beam equation, with the additional consideration of in-plane compressive forces acting on the ice. Employing linear wave theory, the problem is presented as an initial boundary value problem and is solved using Laplace and Fourier transforms to obtain a mathematical expression for the ice sheet's deflection in terms of infinite integrals. These integrals are thereafter solved asymptotically for large time and distance using the stationary phase method. The asymptotic analysis indicates a growing response over time when the poles and stationary points of the phase functions coalesce. The deflection of the floating ice sheet is graphically presented, highlighting the effects of various non-dimensional parameters, such as flexural rigidity, compressive force, uniform current speed, and the angular frequency of the oscillatory pressure. Additionally, the group velocity and phase velocity of flexural gravity waves are derived from the dispersion relation and elucidated using diagrams. The results show that compressive force and current speed significantly influence wave amplitude, enhancing the oscillatory nature, while the flexural rigidity of the elastic plate and the angular frequency of the applied pressure have a substantial impact on the plate's deflection.

Six Points Implicit Block Method with Extra Derivative for Solving Second Order Ordinary Differential Equations

The construction of the six-point block methods with extra derivatives for solving directly is presented in this paper. The suggested block methods concurrently approximate the problem's solution at six points and are formulated using the Hermite Interpolating Polynomial. The block methods do not reduce the equation to a first-order system of initial value problems (IVPs); instead, they directly obtain the numerical solutions. Additionally, the order and zero-stability of the suggested techniques are examined. We present numerical results and draw comparisons with other block methods that are currently in use. The results demonstrate how effective the suggested approaches are at solving second-order IVPs in general.

Legendre Operational Differential Matrix for Solving Fuzzy Differential Equations with Trapezoidal Fuzzy Function Coefficients

In this work, we used the Legendre operational differential matrix method based on Tau method to obtain the fuzzy approximate-analytical solutions of the fuzzy differential equations in which the coefficients are trapezoidal fuzzy function. This method allows for the fuzzy solution of the fuzzy initial ( or boundary) value problems to be computed in the form of an infinite fuzzy series .Also, this method enables to approximate the fuzzy exact-analytical solutions with high efficiency, as these solutions can be resorted to if it is not possible to find the exact solutions of these fuzzy problems. We introduced a comparison between the approximate solutions that we computed and the exact solutions of the chosen problem, as we found the absolute error. According to the numerical results, the series solutions that we found are accurate solutions and very close to the exact solutions.

Improved growth estimate for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity

This paper provides an improved exponential growth estimate, surpassing the growth rate given in the previous work. This finding elucidates the impact of the power index k in the logarithmic nonlinearity uln|u|k on the growth behavior of the solution to the initial boundary value problem for the one-dimensional sixth-order nonlinear Boussinesq equation with logarithmic nonlinearity.

Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework

We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics equations in the half-plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform L∞ estimates of the error functions comes from the unboundedness of vorticity of strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential component of magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl boundary layer ansatz in L∞ sense in Sobolev framework. Compared with the homogeneous incompressible case considered in [33], there exists a strong boundary layer of density. Consequently, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.

Three-dimensional continuum point cloud method for large deformation and its verification

This study presents a strong form based meshfree collocation method, which is named Continuum Point Cloud Method, to solve nonlinear field equations derived from classical mechanics for deformed bodies in three-dimensional Euclidean space. The method and its implementation are benchmarked against a nonlinear vector field using manufactured solutions. The analysis of mechanical fields firstly focuses on the study of St. Venant Kirchhoff and compressible neo-Hookean materials. Results for various initial boundary value problems are presented, including benchmark cases involving unidirectional tension and simple shear. Subsequently, the study concludes with an analysis of a displacement-controlled simulation of a compressible neo-Hookean material, specifically a bar that is pulled to 50% of its original length and rotated 90°. The pure tension case yields a 1.5% error in displacement between computed and expected values and a combined tension and torsion loading case provides further insight into material behavior under complex loading conditions. The resulting normal axial and transverse stress-strain curves are also presented. Finally, the consistency and robustness of the proposed nonlinear numerical schemes are successfully demonstrated through various numerical experiments.

An efficient recursive technique with Padé approximation for a kind of Lane–Emden type equations emerging in various physical phenomena

The study numerically examined a class of nonlinear singular differential problems known as the Lane–Emden differential equation, which emerges in numerous real-world situations. The primary goal of this work is to formulate a computationally efficient iterative technique for solving the nonlinear Lane–Emden initial value problems. The proposed approach is a hybrid of the homotopy perturbation method and the Padé approximation. The nonlinear singular Lane–Emden initial value problem (SLEIVP) is transformed into an equivalent recursive integral employing the Picard’s approach. To resolve the singularity and nonlinearity, the recursive integral equation is transformed into a system of integral equations by using the homotopy notion. Furthermore, to enhance the convergence rate of the technique, Padé approximation is taken into account. The convergence analysis for the proposed approach is also conducted. The present technique is tested on SLEIVPs and numerical findings are compared with the existing techniques, to demonstrate the accuracy, effectiveness and ease of use.

From Lévy walks to fractional material derivative: Pointwise representation and a numerical scheme

The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of Lévy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier–Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation in time and space, pointwise, of the fractional material derivative. This allows us to define it on a space of locally integrable functions which is larger than the original one in which Fourier and Laplace transform exist as functions.We consider several typical differential equations involving the fractional material derivative and provide conditions for their solutions to exist. In some cases, the analytical solution can be found. For the general initial value problem, we devise a finite volume method and prove its stability, convergence, and conservation of probability. Numerical illustrations verify our analytical findings. Moreover, our numerical experiments show superiority in the computation time of the proposed numerical scheme over a Monte Carlo method applied to the problem of probability density function’s derivation.

Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces

In this article, we deal with the initial boundary value problem for a viscoelastic system related to the quasilinear parabolic equation with nonlinear boundary source term on a manifold $\mathbb{M}$ with corner singularities. We prove that, under certain conditions on relaxation function $g$, any solution $u$ in the corner-Sobolev space $\mathcal{H}^{1,(\frac{N-1}{2},\frac{N}{2})}_{\partial^{0}\mathbb{M}}(\mathbb{M})$ blows up in finite time. The estimates of the life-span of solutions are also given.

Initial Boundary Value Problem for the Coupled Kundu Equations on the Half-Line

In this article, the coupled Kundu equations are analyzed using the Fokas unified method on the half-line. We resolve a Riemann–Hilbert (RH) problem in order to illustrate the representation of the potential function in the coupled Kundu equations. The jump matrix is obtained from the spectral matrix, which is determined according to the initial value data and the boundary value data. The findings indicate that these spectral functions exhibit interdependence rather than being mutually independent, and adhere to a global relation while being connected by a compatibility condition.

The Ulam Stability of High-Order Variable-Order φ-Hilfer Fractional Implicit Integro-Differential Equations

This study investigates the initial value problem of high-order variable-order φ-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions.

On the stability of fully nonlinear hydraulic-fall solutions to the forced water wave problem

Two-dimensional free-surface flow over localised topography is examined, with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a dip, respectively. Steady hydraulic-fall solutions to the full incompressible, irrotational Euler equations are computed, and their linear and nonlinear stability is analysed by computing eigenspectra of the pertinent linearised operator and by solving an initial value problem. The computations are carried out numerically using a specially developed computational framework based on the finite-element method. The Hamiltonian structure of the problem is demonstrated, and stability is determined by computing eigenspectra of the pertinent linearised operator. It is found that a hydraulic-fall flow over a bump is spectrally stable. The corresponding flow over a dip is found to be linearly unstable. In the latter case, time-dependent simulations show that ultimately, the flow settles into a time-periodic motion that corresponds to an invariant solution in an appropriately defined phase space. Physically, the solution consists of a localised large-amplitude wave that pulsates above the dip while simultaneously emitting nonlinear cnoidal waves in the upstream direction and multi-harmonic linear waves in the downstream direction.

Solving hyperchaotic initial value problems using second derivative extended trapezoidal rule of the second kind blended with generalized backward differentiation formulae with variable step size

This paper aims to evaluate the accuracy and performance of the second derivative block extended trapezoidal rule of the second kind, blended with generalized backward differentiation formulae using variable step sizes, in solving hyperchaotic initial value problems. Hyperchaotic systems, known for their rapidly changing solutions and extreme sensitivity to initial conditions, pose significant challenges for numerical methods. Traditional approaches often involve linearizing the systems or solving them in multiple sub-intervals, followed by combining the results. However, these methods can introduce errors due to the subdivision of intervals. The proposed method, which integrates the second derivative block extended trapezoidal rule with generalized backward differentiation formulae, offers a more precise solution by eliminating the need for interval subdivision and direct linearization. The continuous formulation of this method is derived using a multistep inversion technique that blends two linear multistep methods, while the discrete schemes are deduced from their continuous interpolants. Convergence analyses of these schemes are presented, demonstrating the efficiency and accuracy of the proposed block methods. As a result, these methods are well-suited for solving hyperchaotic and other nonlinear ordinary differential equations.

Conversion of images of 3D friction maps to study the coupling between coefficient of friction, velocity and contact temperature of the disc brake

The coefficient of friction during braking is not constant. Determining its variability requires measurements in real conditions. This paper presents a methodology of conversion of the measured time profiles of the sliding velocity Veq, contact temperature T, and coefficient of friction f, into analytical formulation of two variables z = f(x,y). This was executed in the following steps: 1) creating a three-column table Veq, T, f from experimental measurements; 2) interpolation using Kriging; 3) creating a contour map of the coefficient of friction dependent on the sliding velocity and temperature, 4) import of a grey scale image of the friction map into the finite element (FE) software, and 5) definition of a function z = f(x,y). The studies were carried out for five composite organic friction materials and cast-iron brake disc of a railway vehicle. The developed five friction maps for each friction pair were based the data from the full-scale bench tests during braking from the initial vehicle velocity of 80 kmh−1, 120 kmh−1, 140 kmh−1, 160 kmh−1, 200 kmh−1 to a stop. In the second part of the study the initial-value problem for the equation of motion combined with the heat conduction problem were formulated and solved numerically using special functionalities of the FE software, e.g. Deformed Geometry, Mathematics. Changes in the vehicle velocity and time profiles of the coefficient of friction were determined from the solution of the coupled thermal problem. The computational FE model was verified by comparing the measured and calculated changes in the vehicle velocity, coefficient of friction and evolutions of the brake disc temperature during single braking. The maximum calculated temperature 1 mm under the contact surface of the brake disc, during braking from initial vehicle velocity of 80 kmh−1, was obtained for the base friction material and was equal to T1,3,5Num.= 72 °C. The corresponding experimental value was T1-6Exp.= 66.1 °C. When braking from 140 kmh−1, for the same material it was T1,3,5Num.= 129.7 °C and T1-6Exp.= 130.7 °C. The obtained differences in braking times calculated and measured did not exceed 1.7 %. The corresponding temperature differences of the disc at the stopping time were lower than 5.5 %.

Operational Calculus for the 1st-Level General Fractional Derivatives and Its Applications

The 1st-level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann–Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the fractional calculus literature. In this paper, we first construct an operational calculus of the Mikusiński type for the 1st-level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann–Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for the derivation of the closed-form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st-level GFDs.

Projection and modified projection methods for nonlinear Hammerstein integral equations on the real line using Hermite polynomials

Many physical problems represented as initial and boundary value problems are usually solved by transforming them into integral equations on the real line. Therefore, this paper proposes polynomially based projection and modified projection methods to solve Hammerstein integral equations on the real line with sufficiently smooth kernels. The approximating operator employed is either the orthogonal projection or an interpolatory projection using Hermite polynomials as basis functions. We analyse the convergence of the proposed approaches and its iterated version and we establish superconvergence results. Through different numerical tests, the effectiveness of the proposed methods is presented to demonstrate the given theoretical framework.

Some Application for a Fuzzy Differential Equation and Solve by Runge-Kutta Method

In this article, the starting condition was defined using a fuzzy initial value problem (IVP). Additionally, we discussed various methods for solving fuzzy differential equations, including the modified two-step Simpson method and Runge-Kutta of orders (two, three, four, five, and six). For each method, we provided a numerical example and the known convergence rates of the solutions. Then we discussed the comparison of the solutions of all methods, using computer software to offer rough solutions for the Runge Kutta method. And take some application solve by Runge-Kutta in physics and medical

Nonlocal Extensions of First Order Initial Value Problems

Nonlocal Extensions of First Order Initial Value Problems

Non-uniform dependence on initial data for the Euler equations in Besov spaces

In this paper, we consider the initial value problem to the higher dimensional Euler equations in the whole space. Based on the local well-posedness result and the lifespan, we prove that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces. Our obtained result improves considerably the previous results given by Himonas-Misiołek (2010) [8] and Pastrana (2021) [21].