Search Results (829)

This paper presents a comparative study of two approaches for solving second-order ordinary differential equations arising from the membrane theory of shells of revolution. The considered equations originate from the rotational symmetry of shell structures, which enables the reduction of the governing partial differential equations to a sequence of ordinary differential equations corresponding to individual circumferential harmonics. The study compares the classical Differential Transform Method (DTM) with Physics-Informed Neural Networks (PINNs). Both initial value and boundary value problems are investigated, including benchmark examples with known analytical solutions and a systematic analysis of the influence of PINN architecture on the solution accuracy. For the PINN approach, the effects of the number of collocation points, hidden layers, and neurons per layer on the approximation error and training time are examined. The results demonstrate that DTM provides an efficient framework for constructing analytical solutions of initial value problems with minimal computational cost. However, its application to boundary value problems requires the introduction of additional auxiliary parameters and the solution of supplementary nonlinear equations, considerably increasing the analytical complexity of the procedure. In contrast, PINNs achieve high accuracy for both initial and boundary value problems while naturally incorporating boundary conditions through the loss function. The presented results demonstrate how the exploitation of geometric symmetry, combined with modern scientific machine learning techniques, provides an effective computational framework for solving differential equations arising in shell mechanics. Full article

This paper presents a numerical method for solving transient one-dimensional heat conduction problems in multilayer slabs with temperature-dependent thermal conductivities. The governing nonlinear partial differential equations are formulated separately in each layer, allowing for distinct material properties. Perfect thermal contact at internal interfaces is enforced through continuity of temperature and heat flux, while general boundary conditions are imposed at the external boundaries, including prescribed temperature, specified heat flux, and convective exchange. A key feature of the proposed approach is to discretize the partial differential equations first in time using the implicit Euler method, thereby reducing the original problem to a sequence of nonlinear two-point boundary value problems with interface (transmission) conditions. A second-order finite difference scheme is employed for spatial discretization, and the resulting system is expressed in global form using a unified indexing strategy. The system is solved at each time step by Newton linearization, yielding a sparse Jacobian matrix that is tridiagonal in the interior and locally extended at the interfaces. Efficient banded solvers lead to O(N) cost per time step, where N is the number of spatial nodes. Numerical experiments confirm the expected accuracy, unconditional stability, and computational complexity of the method. Full article

Three-dimensional Riemann–Liouville time-fractional integro-differential equations provide useful prototype models for diffusion and transport processes with temporal memory and weakly singular hereditary effects. Their numerical solution is challenging because of the nonlocal fractional derivative, history-dependent fractional integral term, and large-scale discrete systems arising from three-dimensional spatial discretization. In this work, an efficient high-order compact finite difference scheme is developed for solving such problems. The Riemann–Liouville fractional derivative is approximated by the weighted and shifted Grünwald difference formula, the fractional integral term is discretized by the product trapezoidal formula, and the Laplace operator is approximated by compact difference operators. The proposed scheme achieves second-order accuracy in time and fourth-order accuracy in space. Moreover, the solvability, stability, and convergence of the fully discrete three-dimensional scheme are analyzed under suitable regularity assumptions. Numerical experiments, including examples with smooth and non-smooth solutions, verify the theoretical convergence orders and demonstrate the effectiveness of the proposed method for different fractional parameters. Full article

We study an optimal control framework for chemotherapy administration in patients with chronic lymphocytic leukemia (CLL) while accounting for immune regulation and treatment-induced allergic reactions. The analysis is based on a previously developed nonlinear delay differential equation model describing the interactions between leukemic cells, immune populations, antigen-presenting cells, and cytokine dynamics, with three distinct biological delays. The chemotherapy infusion rate is introduced as a time-dependent control variable and optimized to reduce leukemic burden, shift the helper T-cell balance toward a Th1-dominant configuration associated with lower hypersensitivity risk, and preserve immune competence. Existence of an optimal control is established for arbitrary delays and horizon, without the commensurability hypothesis required by reductions in delay systems to higher-dimensional delay-free ones; the argument uses only that the control enters the dynamics affinely and the running cost concavely. Necessary optimality conditions are derived via Pontryagin’s Maximum Principle for systems with delays, and the resulting eleven-dimensional adjoint system, which carries advanced arguments generated by the three delays, is written out explicitly. A contraction estimate for the associated sweep operator yields both uniqueness of the optimal control on a short horizon and geometric convergence of the numerical scheme. The optimality system is solved by a forward–backward sweep adapted to the delayed setting, with documented convergence and grid independence and sensitivity analysis over kinetic parameters, delays, initial conditions and objective weights. The optimized schedule is compared not only with the untreated case and a low constant dose, but also with a constant infusion delivering the same cumulative exposure, so that the reported benefit is attributable to the temporal distribution of the dose rather than to its total amount. At equal exposure, the optimal schedule reaches each therapeutic milestone earlier—Th1 dominance 0.9 days sooner and a 90% leukemic reduction 1.6 days sooner—and attains a terminal leukemic burden lower by a factor of 2.25; a constant infusion of the same total dose reaches a comparable configuration later. The benefit of adaptive scheduling in this model is therefore principally one of rate of response at fixed drug exposure. We emphasize that the absolute Th2 population is not reduced by treatment; the reduction in hypersensitivity risk arises from the resulting Th1-dominant relative balance rather than from direct Th2 suppression. To characterize the therapeutic outcome in information-theoretic terms, we describe the two competing goals as distributional balances: an allergy axis, given by the Th1/Th2/Treg distribution, and a leukemia axis, given by the immune/leukemic distribution, each measured by its Shannon entropy and its Kullback–Leibler divergence to a healthy reference profile. These quantities are used in two roles. As diagnostics, they are evaluated along the computed trajectories, and the ordering of dosing strategies is shown to be robust across twenty alternative reference profiles. As an objective, the combined divergence is then taken as the running cost of a second optimal control problem; because it depends on the leukemic population only through a normalized fraction, it prescribes a markedly gentler schedule that administers 37% of the drug and still achieves a 93% leukemic reduction, against 98% for the population-based formulation. These results suggest that adaptive, immune-aware chemotherapy scheduling may accelerate disease control at fixed drug exposure, and that information-theoretic objectives offer a scale-free alternative formulation of the therapeutic goal. Full article

Delay differential equations form a distinct class of differential equations in which the derivative of the dependent variable depends on its value at an earlier time. This unique structure makes them particularly suitable for modeling phenomena in areas such as population dynamics, epidemiology, and the spread or control of diseases. In this study, a high-order Haar wavelet collocation method (HoHWCM) is proposed for the numerical solution of second-order delay differential equations (SoDDEs). The delay term is approximated using a Taylor expansion, transforming the SoDDE into a standard second-order differential equation (SoDE). The nonlinear terms are linearized through an innovative Taylor series-based approach, which also serves as an efficient iterative scheme. The resulting SoDE is then discretized using Haar wavelet basis functions, yielding a system of linear algebraic equations that is solved iteratively. This strategy eliminates the need for Newton’s or Broyden’s methods, thereby reducing computational cost and improving time efficiency. A variety of linear and nonlinear benchmark problems are solved to evaluate the accuracy and efficiency of the proposed method. Comparative results with established approaches from the literature demonstrate that the HoHWCM achieves higher accuracy, faster convergence, and reduced computational time, making it a highly effective alternative for solving such problems. Full article

We study equations $y^{″}+P\left(x\right)y^{\prime }+Q\left(x\right)y=0$, with $P,Q\in \mathbb{C}\left(x\right)$, that admit two linearly independent local solutions $y_1=x^{r}A\left(x\right)$ and $y_2=x^{s}B\left(x\right)$ at the regular singular point 0, where A and B are polynomials and $A\left(0\right)B\left(0\right)\ne 0$. We call these Frobenius-polynomial type ($\mathrm{FP}$) equations and focus on the subclass for which both roots of the indicial equation at $x=0$ are real. Writing $W(y_1,y_2)=x^{r+s-1}R\left(x\right)$ and $W(y_1^{\prime },y_2^{\prime })=x^{r+s-3}{R}_1\left(x\right)$, we derive $P\left(x\right)=-(r+s-1)/x-R^{\prime }\left(x\right)/R\left(x\right)$ and $Q\left(x\right)=R_1\left(x\right)/\left(x^{2}R\left(x\right)\right)$. Factoring R determines every pole and residue of P, while its vanishing order at 0 distinguishes the possible exponent representations. We prove that the degree equation at ∞ has two distinct real roots. For each solution exponent, subtracting that exponent from the two roots leaves at most two non-negative-integer candidate degrees for its polynomial factor. For proper rational functions P and Q whose coefficients are algebraic over $\mathbb{Q}$, we give a recognition algorithm and prove its termination, soundness, and completeness for the real-indicial subclass, including the resonant case. The final step tests the existence and, in the common-exponent case, the linear independence of the required polynomial solutions by solving finitely many homogeneous linear systems for their coefficients. Three Calogero-type families, a rational example outside those families, and a negative general-Heun-type example illustrate the procedure. Full article

Nonlocal circumstances in genetic engineering are crucial as they pertain to the understanding of genetic material. When these conditions are associated with differential integral equations, particularly concerning the time variable, they yield comprehensive insights into the material’s temporal memory, which can be advantageous for understanding all material properties (including chronic conditions or behavioral characteristics), thereby assisting specialists in managing its future evolution. The novelty of this manuscript resides in the exploration of fractional nonlinear mixed integro-differential equations (FrN-MIo-DE) under nonlocal conditions, employing the Toeplitz matrix method with a genetic application. This issue has previously been examined via the Nyström technique and solely under local conditions. A category of mathematical problems prevalent in many domains, including physics, engineering, and biological systems, is fractional calculus. Fractional calculus, which generalizes classical differentiation and integration to non-integer orders, offers a robust foundation for modeling memory and hereditary characteristics in complex systems. We examine the existence and uniqueness of solutions to FrNMIo-DE under nonlocal restrictions, using a discontinuous kernel dependent on location and time-space $L_2[-\mathrm{1,1}]\times C[0,T]$, where T < 1, via analytical methods. According to the features of fractional integrals, FrNMIo-DE adheres to the second-kind Volterra–Hammerstein integral equation (V-HIE), characterized by a discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time for the Volterra integral (VI) term. Subsequently, we use a separation approach technique to produce HIE with time-dependent physical coefficients. Following an analysis of the system’s convergence, a nonlinear algebraic system (NAS) is constructed using the Toeplitz matrix technique (TMT) and related methodologies. The numerical data and associated errors are shown via the Maple 2022 software. Full article

In this paper, we introduce the Appell–Hermite–Fibonacci polynomials within the framework of Fibonomial calculus, combining the Appell structure with a Hermite-type deformation governed by Fibonacci coefficients. The family is defined through an appropriate F-exponential generating function, from which its principal properties naturally follow. Explicit representations, including convolution identities and series expansions, are established. A determinantal formulation preserving the lower–Hessenberg structure of Fibonacci–Appell systems is obtained, together with a matrix realization linked to a generalized Fibonacci–Pascal matrix. The operational framework yields lowering and raising relations, a second-order F-differential equation, and a Rodrigues-type representation. Under suitable conditions, orthogonality properties are also derived. The results place this hybrid family within a coherent extension of Appell theory in the Fibonacci setting. Full article

A fully coupled three-dimensional (3D) thermo-magneto-elastic layer-wise formulation is developed for the analysis of multilayered flat and curved panels used in aerospace and aeronautical applications. The model relies on a system of coupled second-order differential equations along the thickness coordinate z, formulated in a mixed orthogonal curvilinear reference system. The governing equations combine the three-dimensional equilibrium equations with the magnetic induction divergence equation and the heat conduction equation, providing a unified multifield framework for thermo-magneto-elastic analyses. Through a suitable definition of the curvature parameters, the same formulation can be directly applied to plates, cylinders, cylindrical panels, and shells with constant radii of curvature. The governing equations are analytically solved by adopting harmonic expansions in the in-plane directions together with the exponential matrix method along the thickness coordinate. The harmonic representation naturally satisfies simply-supported boundary conditions along the panel edges. The multilayered structure is modeled according to a layer-wise strategy, where the continuity of the selected mechanical, magnetic, and thermal variables is enforced across the interfaces between adjacent layers. Different loading boundary conditions can be assigned at the external surfaces by prescribing pressure loads, magnetic potential, transverse magnetic induction, and over-temperature. The numerical investigation is divided into two stages. First, the accuracy of the proposed formulation is verified through comparisons with thermo-magneto-elastic solutions available in the literature. Then, a comprehensive set of new benchmark results is presented by considering different geometries, thickness ratios, and loading boundary conditions. Both tabulated values and through-the-thickness distributions are reported for the most significant field variables. These benchmark results provide useful reference data for the assessment and validation of future two-dimensional and three-dimensional analytical and numerical formulations devoted to coupled thermo-magneto-elastic problems. Full article

This work studies an optimal control problem for a wave equation with a moving localized damping, modeled by the characteristic function of a ball whose center evolves according to a controlled second-order dynamical system. The objective is to minimize the $H^1$-norm of the time derivative of the wave at a final time, while penalizing the control effort through a quadratic cost on the acceleration. We first establish the local well-posedness of the coupled system, consisting of a damped wave equation and an ordinary differential equation for the damping center. Under suitable compactness and regularity assumptions on the admissible controls, we prove the existence of at least one optimal control via the direct method of the calculus of variations. We then derive first-order necessary optimality conditions through an extended Lagrangian formalism, yielding a coupled system involving the primal state, two adjoint states, and a pointwise relation linking the optimal control to the adjoint variable. The gradient of the cost functional is computed explicitly, showing that the sensitivity is concentrated on the boundary of the moving damping zone. Finally, a numerical implementation based on a gradient descent algorithm is proposed, using a Newmark scheme for the wave equation and a Verlet scheme for the trajectory, with regularization of the nonsmooth indicator function. The resulting algorithm provides a systematic approach for computing optimal damping trajectories in applications such as vibration suppression and noise control. Full article

We study a nonlocal boundary value problem for a system of functional-differential equations with involution and an additional parameter. The proposed approach is based on a parameterization method developed by D. Dzhumabaev. The original boundary value problem is transformed into an equivalent Cauchy problem posed at the midpoint of the interval, together with a system of algebraic equations for the unknown parameters. By exploiting the symmetry and antisymmetry properties of the solution, we reduce the resulting Cauchy problem to a system of coupled Volterra integral equations of the second kind. This reduction makes it possible to derive explicit solution representations and to establish necessary and sufficient conditions for unique solvability in terms of the invertibility of the matrix generated by the boundary conditions. Finally, a first-order functional-differential equation is analyzed to demonstrate the applicability of the proposed method and to illustrate the obtained solvability conditions. Full article

The present work introduces a fully resolved three-dimensional thermal Lattice Boltzmann framework developed to investigate the impact of freely moving, finite-size spherical particles on natural convection within a cubic enclosure. The fluid-phase momentum and energy fields are resolved using coupled double-distribution function kinetic approach, while the solid phase is governed by explicitly coupled linear, angular, and thermal conservation equations. To accurately map the moving spherical surfaces onto the Eulerian lattice grid, a second-order linear interpolated bounce-back scheme is implemented. The conjugate heat transfer between the phases is simplified via a lumped capacitance model, assuming negligible internal thermal resistance within the solid spheres. Short-range particle–particle and particle–wall interactions are handled using Glowinski’s repulsive force model. The spatial accuracy of the framework is validated using a circular Taylor–Couette flow benchmark—demonstrating second-order spatial convergence and a differentially heated natural convection in a cubic cavity benchmark, yielding bulk Nusselt numbers within $1\%$ of established literature data. This validated tool is subsequently used to analyze the complex interplay between particulate motion and bulk thermal transport efficiency. Analysis of the temperature fields reveals that the overall thermal structure is governed primarily by the Rayleigh number, while the low particle concentration produces only minor modifications to the convective heat transfer. In contrast, the particle distribution exhibits a strong dependence on the flow intensity. Full article

An adaptive immersed boundary method (IBM) for simulating non-isothermal incompressible flows with convective heat transfer involving a rotating circular cylinder is developed. Both Dirichlet- (isothermal) and Neumann (zero heat flux)-type temperature boundary conditions are implemented. In addition to the discrete momentum forcing and energy forcing adopted to effectively satisfy the prescribed velocity and temperature boundary conditions, a mass source/sink term is introduced into the continuity equation to meet the mass conservation at the immersed boundary. The Navier–Stokes equations are solved using the fractional step method implemented on a staggered Cartesian grid system. Time stepping is performed using a second-order Adams–Bashforth/backward-differentiation method, while spatial derivatives are approximated with a second-order centered scheme. Testing of the flow induced by a rotating disk demonstrates that the spatial accuracy of the presented algorithm is second-order. Furthermore, the proposed method is validated by forced convective flow past a rotating isothermal circular cylinder. Finally, mixed Rayleigh–Bénard convection in a square cavity with an embedded adiabatic rotating circular cylinder is simulated, showing that heat transport can be greatly enhanced by increasing the rotating rate and radius of the cylinder at larger Prandtl numbers in the laminar regime. Full article

In this study, we develop a novel variable-order fractional extension of an eight-dimensional (8D) hyperchaotic differential equation system modeled via the Liouville–Caputo operator. Moving beyond constant fractional-order models, our system implements time-variable orders, which enable its historical memory structure to evolve dynamically over time. To numerically approximate the trajectories of this complex 8D system, a second-order Lagrange numerical integration approach is formulated. An extensive dynamic analysis explores the behavior of this variable-order framework under two distinct configurations: a slowly periodic memory function and a smooth, monotonic hyperbolic tangent function. Topological complexity and multidimensional chaos are characterized using parameter-dependent bifurcation diagrams, phase portraits, Kaplan–Yorke fractal dimensions, and Kolmogorov–Sinai metric entropy. Numerical results show that both variable-order configurations display robust hyperchaotic dynamics characterized by four positive Lyapunov exponents. Crucially, the proposed variable-order extension enhances the phase space footprint of the baseline system, achieving a maximum Kaplan–Yorke dimension of 7.100, thereby offering excellent topological density for secure cryptographic applications. Full article

Many applications in finance and biology involve multiplicative noise geometric Brownian motion (GBM)-type stochastic differential equations (SDEs) whose drift carries a single dominant periodic component. Such structures arise in seasonal Black–Scholes option pricing, commodity derivatives with annual price cycles, and stochastic biological oscillators driven by a known frequency; the primary contribution of this paper is a high-precision numerical scheme validated on GBM-type test problems with periodic drift. This paper proposes a Strang operator splitting scheme within the Logarithmic Drift-Diffusion Splitting (LDDS) framework, which splits the SDE in y-space into a deterministic drift ODE sub-step (Step A) and an exactly solvable multiplicative diffusion sub-step (Step B). Step A employs the Periodically Fitted Adams–Bashforth–Moulton fourth-order predictor–corrector method (PABM4), which achieves zero local truncation error for trigonometric forcing terms by introducing additional shift terms and simultaneously imposing polynomial exactness conditions and trigonometric fitting conditions. When the Step A forcing belongs to the PABM4 exact function class ${\mathcal{F}}_{\omega }=\mathrm{span}\{1,t,t^2,\sin \omega t,\cos \omega t\}$, the Strang+PABM4 scheme achieves floating-point precision saturation. We investigate three test problems: the cosine-drift GBM (Test Problem 1), the polynomial–trigonometric mixed drift GBM (Test Problem 2), and a dual-frequency drift applicability test (Test Problem 3). Monte Carlo strong error experiments ($M=1000$ paths) validate that Strang+PABM4 achieves saturation at machine precision (≈${10}^{-15}$) on Test Problems 1 and 2, improving precision by ≈${10}^2\times$ over the best algebraically convergent reference. Test Problem 3 identifies the method’s applicability boundary: when the drift contains a second frequency outside ${\mathcal{F}}_{\omega }$, Strang+PABM4 degrades gracefully to order ≈ 4 without catastrophic failure. The floating-point saturation of Strang+PABM4 is contingent on the drift belonging to ${\widehat{\mathcal{F}}}_{\omega }$; when this condition is violated, the method degrades gracefully to algebraic order ≈ 4, as demonstrated in Test Problem 3. Full article