Initial Value Problem Research Articles

A New Fractional Approach for the Higher-Order q-Taylor Method

The main goal of this work is to propose a new fractional approach of the higher order q-Taylor method with Initial Value Problems (IVPs) for fractional q-difference equations which is called the Fractional Higher-Order q-Taylor Method (FHOqTM). By applying the generalised q-Taylor theorem, this would be achieved. As a consequence, we calculate the FHOqTM’s local truncation error. Finally, we present numerical applications to validate our results by comparing the exact solution and the approximatesolution resulting from (FHOqTM).

Applications of Catalan Neural Network (CaNN) model

In this paper, we construct Catalan neural network (CaNN) model and examine the nonlinear solvers of the ordinary nonlinear differential equation using the CaNN model with linear and nonlinear activation functions. Using a single-layer functional linking neural network (FLANN) architecture, we extend input models using the first five Catalan polynomials, update network parameters that we initially randomly select, and use error backpropagation algorithms. The CaNN trial solution for solving nonlinear differential equations is in excellent agreement with the analytical solution. Also, the validity of the CaNN method is investigated by treating second-order differential equations of Lane–Emden type as boundary and initial value problems.

Piecewise Neural Network Method for Solving Large Interval Solutions to Initial Value Problems of Ordinary Differential Equations

Piecewise Neural Network Method for Solving Large Interval Solutions to Initial Value Problems of Ordinary Differential Equations

Qualitative analysis of fourth-order hyperbolic equations

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations.

A shooting-Newton procedure for solving fractional terminal value problems

In this paper we consider the numerical solution of fractional terminal value problems: namely, terminal value problems for fractional differential equations. In particular, the proposed method uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems, i.e., initial value problems for fractional differential equations. As a result, the method is able to produce spectrally accurate solutions of fractional terminal value problems. Some numerical tests are reported to make evidence of its effectiveness.

Heat and mass flux dynamics of tangent hyperbolic nanofluid flow with unsteady rotatory stretching disk over Darcy-Forchheimer porous medium

Abstract The purpose of the research is to examine a tangent hyperbolic nanofluid flowing in three dimensions (3D) axisymmetrically on an unsteady rotatory stretching disk over a Darcy-Forchheimer porous medium. First order initial value problems (IVPs) are generated from the governing partial differential equations (PDEs) through the use of similarity transformation and linearization. The Runge-Kutta sixth order (RK6) is utilized to solve the IVP system using the shooting technique and the built-in Python program ‘fsolve model10’. Articles that have already been published are used to validate the implemented approach. Graphs are used to examine how various parameters affect velocity, temperature, and concentration. Additionally, the behavior of heat, mass flux, and skin friction in response to different parameters is investigated. The study’s findings showed that as the Forchheimer number and velocity slip parameter increased, the nanofluid’s radial and tangential velocities decreased as well. As temperature and concentration slip parameters increase, correspondingly, thicker and thinner boundary layer structures are seen. The drag force in the tangential and radial direction behaves in the same manner. Both the rates of heat and mass transfers are initiated for an increase Eckert and Prandtl numbers and demotivated for power-law index number. The dissipation effect with radiation and chemical reaction plays a major role in heat and mass fluxes, respectively. The study can be used in various computer storage, coatings, lubricants, and coolants.

Numerical Solutions of Volterra Integral Equations with Subdivision-Based Collocation Approach

Integral equations play a vital role in the fields of applied and computational mathematics. This paper introduces a unique subdivision-based collocation technique for solving the initial value problems (IVPs) arising from Volterra integral equations (VIEs). A suitable subdivision scheme is employed to develop the numerical algorithm for solving the VIEs. Several numerical illustrations are tested and compared with the existing methods for assessing the efficiency of the proposed numerical algorithm and error estimation.

Step-by-step time discrete Physics-Informed Neural Networks with application to a sustainability PDE model

The use of Artificial Neural Networks (ANNs) has spread massively in several research fields. Among the various applications, ANNs have been exploited for the solution of Partial Differential Equations (PDEs). In this context, the so-called Physics-Informed Neural Networks (PINNs) are considered, i.e. neural networks generally constructed in such a way as to compute a continuous approximation in time and space of the exact solution of a PDE.In this manuscript, we propose a new step-by-step approach that allows to define PINNs capable of providing numerical solutions of PDEs that are discrete in time and continuous in space. This is done by establishing connections between the network outputs and the numerical approximations computed by a classical one-stage method for stiff Initial Value Problems (IVPs). Links are also highlighted between the step-by-step PINNs derived here, and the time discrete models based on Runge–Kutta (RK) methods proposed so far in literature. To evaluate the efficiency of the new approach, we build such PINNs to solve a nonlinear diffusion-reaction PDE model describing the process of production of renewable energy through dye-sensitized solar cells. The numerical experiments show that not only the new step-by-step PINNs are able to well reproduce the model solution, but also highlight that the proposed approach can constitute an improvement over existing continuous and time discrete models.

Numerical solutions for second-order neutral volterra integro-differential equations: Stability analysis and finite difference method

This work deals with the initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established. In support of theoretical results, numerical results are performed by employing the proposed numerical technique.

Relaxation of first-class constraints and the quantization of gauge theories: From “matter without matter” to the reappearance of time in quantum gravity

We make a conceptual overview of a particular approach to the initial-value problem in canonical gauge theories. We stress how the first-class phase-space constraints may be relaxed if we interpret them as fixing the values of new degrees of freedom. This idea goes back to Fock and Stueckelberg, leading to restrictions of the gauge symmetry of a theory, and it corresponds, in certain cases, to promoting constants of Nature to physical fields. Recently, different versions of this formulation have gained considerable attention in the literature, with several independent iterations, particularly in classical and quantum descriptions of gravity, cosmology, and electromagnetism. In particular, in the case of canonical quantum gravity, the Fock–Stueckelberg approach is relevant to the so-called problem of time. Our overview recalls and generalizes the work of Fock and Stueckelberg and its physical interpretation with the aim of conceptually unifying the different iterations of the idea that appear in the literature and of motivating further research.

On a Hybrid Class of $p$-Laplacian Initial Value Problems with Modified Mittag-Leffler Kernel

In this work, we establish key results on the existence theory for a category of initial value problems (IVPs) involving hybrid fractional integro-differential equations (HFIDEs) with a $p$-Laplacian operator, utilizing the modified Mittag-Leffler kernel. By employing Krasnoselskii and Banach fixed point theorems (FPTs), we determine the conditions required for the existence of solutions. Additionally, we examine the Hyers-Ulam (H-U) stability of the problem. Lastly, we present an example to confirm our theoretical results.

Inverse Problem for the Moisture Transfer Equation: Development of a Method for Finding the Unknown Parameter and Proof of the Convergence of the Iterative Process

This study develops a mathematical model for soil moisture diffusion, addressing the inverse problem of determining both the diffusion coefficient and the variation coefficient in a nonlinear moisture transfer equation. The model incorporates specific boundary and initial conditions and utilizes experimentally measured moisture values at a boundary point as input data. An iterative method, based on an explicit gradient scheme, is introduced to estimate the soil parameters. The initial boundary value problem is discretized, leading to a difference analog and the formulation of a conjugate difference problem. Iterative formulas for calculating the unknown parameters are derived, with a priori estimates ensuring the convergence of the iterative process. Additionally, the research establishes the convergence of the numerical model itself, providing a rigorous foundation for the proposed approach. The study also emphasizes symmetry in moisture calculations, ensuring consistency regardless of the calculation direction (from right to left or left to right) and confirming that moisture distribution remains symmetric within specified intervals. This preservation of symmetry enhances the model’s robustness and accuracy in parameter estimation. The numerical simulations were successfully conducted over a 7-day period, demonstrating the model’s reliability. The discrepancy between the numerical predictions and experimental observations remained within the margin of measurement error, confirming the model’s accuracy.

Ulam–Hyers and Generalized Ulam–Hyers Stability of Fractional Differential Equations with Deviating Arguments

In this paper, we study the initial value problem for the fractional differential equation with multiple deviating arguments. By using Krasnoselskii’s fixed point theorem, the conditions of solvability of the problem are obtained. Furthermore, we establish Ulam–Hyers and generalized Ulam–Hyers stability of the fractional functional differential problem. Finally, two examples are presented to illustrate our results, one is with a pantograph-type equation and the other is numerical.

Oscillatory behavior of Ψ$$ \Psi $$‐Hilfer generalized proportional fractional initial value problems

This paper deals with the oscillatory behavior of the ‐Hilfer generalized proportional fractional initial value problem. Using the Volterra integral equation and Young's inequality, we establish sufficient conditions for each solution of the problem to oscillate. For the appropriate choice of the kernel , our obtained results generalize and recover some existing results in the literature. Additionally, we present some examples to emphasize the importance of our results.

A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0, with m−1<α≤m,m∈N. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.

Thermal transportation of radiative cathode nanotube‐based nanofluid on a stretching/shrinking wedge with varying magnetism and heat source

AbstractThe complex heat transfer processes occurring in a magnetized water‐based hybrid nanofluid flowing past a stretching or shrinking wedge are investigated in this work. The research highlights the significant influence of the wedge angle on fluid flow patterns and temperature gradients, emphasizing how larger angles cause changes in pressure, flow separation, and temperature profiles. Additionally, the study emphasizes the importance of radiation effects near surfaces, where increased radiation often leads to cooling due to thermal energy emission. The findings have implications in industries such as aerospace, electronics, thermal management, and groundwater resources management where precise temperature control is critical for safe and efficient operations. The numerical method applied to solve the governing equations in the study is the fourth‐order Runge–Kutta technique with shooting technique. This method efficiently solves the boundary value problem by converting it into a system of first‐order initial value problems. Then shooting technique is used to solve the system, starting with an adequate initial guess. The study's results are presented in tabular and graphical forms, providing valuable insights into the complex fluid dynamics and thermal behavior in the given setting, enabling more informed decision‐making and optimization in relevant engineering applications. Increased radiation frequently results in a cooling effect owing to thermal energy output. Heat sinks/heat sources are crucial in temperature regulation and are effective in lowering temperatures via processes. We also emphasize the importance of radiation impacts near surfaces, where increased radiation frequently leads to cooling owing to thermal energy emission. The numerical values of skin friction and Nusselt number for various parameters in the cases of shrinking and stretching provide crucial insights into the heat transmission and frictional properties of the system under examination. These quantitative results contribute to a better understanding of the complex fluid dynamics and thermal behavior in the given setting, emphasizing the significance of the research findings.

Interaction of driven ``cold'' electron plasma wave with thermal bulk via ion spatial inhomogeneity

Abstract Using high resolution Vlasov - Poisson simulations, evolution of driven ``cold" electron plasma wave (EPW) in the presence of stationary inhomogeneous background of ions is studied. Mode coupling dynamics between ``cold'' EPW with phase velocity $v_{\phi}$ greater than thermal velocity i.e $v_{\phi} \gg v_{thermal}$ and its inhomogeneity induced sidebands is illustrated as an initial value problem. In driven cases, formation of BGK like phase space structures corresponding to sideband modes due to energy exchange from primary mode to bulk particles via wave-wave and wave-particle interactions leading to particle trapping is demonstrated for inhomogeneous plasma. Qualitative comparison studies between initial value perturbation and driven problem is presented, which examines the relative difference in energy transfer time between the interacting modes. Effect of variation in background ion inhomogeneity amplitude as well as ion inhomogeneity scale length on the driven EPWs is reported.

Linear stability analysis of oblique Couette–Poiseuille flows

We perform a detailed numerical study of modal and non-modal stability in oblique Couette–Poiseuille profiles, which are among the simplest examples of three-dimensional boundary layers. Through a comparison with the Orr–Sommerfeld operator for the aligned case, we show how an effective wall speed succinctly characterizes modal stability. Large-scale parameter sweeps reveal that the misalignment between the pressure gradient and wall motion is, in general, destabilizing. For flows that are sufficiently oblique, the instability is found to depend exclusively on the direction of wall motion and not on its speed, a conclusion supported, in part, by the perturbation energy budget and the evolution of the critical layers. Closed forms for the critical parameters in this regime are derived using a simple analysis. From a non-modal perspective, pseudoresonance is examined through the resolvent and the $\epsilon$ -pseudospectra. An analysis of the unforced initial value problem shows that the maximum energy gain is highly dependent on both the magnitude and direction of the wall velocity. However, the strongest amplification is always achieved for configurations that are only weakly skewed. Finally, the optimal perturbations appear to develop via a lift-up effect enhanced by an Orr-like mechanism, the latter driven by cross-flow shear.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.