Search Results (49)

This study introduces a modified computational scheme for handling linear and nonlinear fractal time-dependent partial differential equations. The method is constructed using three different stages that provide third-order accuracy in the fractal time variable. The stability of the approach is examined using scalar fractal models and Fourier analysis, while convergence is established for coupled convection–diffusion systems. The numerical algorithm is applied to analyze the mixed convective flow of a Carreau–Yasuda non-Newtonian fluid over stationary and oscillating plates under the influence of viscous dissipation and magnetic field effects. For spatial discretization, the incompressible continuity equation is handled by a first-order difference scheme, whereas higher-order compact schemes are implemented for the momentum, thermal, and concentration equations. The numerical findings show that increasing the Weissenberg number and magnetic field inclination reduces the velocity distribution. An accuracy assessment against existing numerical techniques demonstrates that the present method yields smaller computational errors, particularly when central difference schemes are used in space. In addition, a surrogate machine learning model is developed to predict the skin friction coefficient and local Nusselt number using Reynolds, Weissenberg, Prandtl, and Eckert numbers as input features. The predictive capability of the model is validated through Parity plots, bar charts for sensitivity analysis, scatter visualization, and Taylor Diagrams, confirming strong agreement with the numerical results. Full article

The Burgers–Huxley equation is a nonlinear partial differential equation that incorporates convective, diffusive and reactive effects and arises in various reaction–diffusion and fluid flow models. In this paper, a numerical method based on the method of lines is proposed for its solution. The spatial derivatives are approximated using a third-order finite difference scheme, which converts the governing partial differential equation into a system of ordinary differential equations. The resulting semi-discrete system is solved in time using the classical fourth-order Runge–Kutta method. The stability and convergence properties of the proposed scheme are analyzed to establish its numerical reliability. Several numerical experiments are presented to illustrate the accuracy and efficiency of the method. The computed results confirm that the proposed approach provides accurate and stable solutions for the Burgers–Huxley equation. Full article

In this work, we develop a disturbance suppression-oriented fuzzy sliding mode secured sampled-data controller for third-order parabolic partial differential equations that ought to cope with nonlinearities, hybrid cyber attacks, and modeled disturbances. This endeavor is mainly driven by formulating an observer model with a T–S fuzzy mode of execution that retrieves the latent state variables of the perceived system. Progressing onward, the disturbance observers are formulated to estimate the modeled disturbances emerging from the exogenous systems. In due course, the information received from the system and disturbance estimators, coupled with the sliding surface, is compiled to fabricate the developed controller. Furthermore, in the realm of security, hybrid cyber attacks are scrutinized through the use of stochastic variables that abide by the Bernoulli distributed white sequence, which combat their unpredictability. Proceeding further in this framework, a set of linear matrix inequality conditions is established that relies on the Lyapunov stability theory. Precisely, the refined looped Lyapunov–Krasovskii functional paradigm, which reflects in the sampling period that is intricately split into non-uniform intervals by leveraging a fractional-order parameter, is deployed. In line with this pursuit, a strictly $({\Phi }_1,{\Phi }_2,{\Phi }_3)-\varrho$ dissipative framework is crafted with the intent to curb norm-bounded disturbances. A simulation-backed numerical example is unveiled in the closing segment to underscore the potency and efficacy of the developed control design technique. Full article

We investigate the Lagrange formulation for the one-dimensional Saint Venant–Exner system. The system describes shallow-water equations with a bed evolution, for which the bedload sediment flux depends on the velocity, $Q\left(t,x\right)=A_g{u}^m,m\ge 1$. In terms of the Lagrange variables, the nonlinear hyperbolic system is reduced to one master third-order nonlinear partial differential equation. We employ Lie’s theory and find the Lie symmetry algebra of this equation. It was found that for an arbitrary parameter m, the master equation possesses four Lie symmetries. However, for $m=3$, there exists an additional symmetry vector. We calculate a one-dimensional optimal system for the Lie algebra of the equation. We apply the latter for the derivation of invariant functions. The invariants are used to reduce the number of the independent variables and write the master equation into an ordinary differential equation. The latter provides similarity solutions. Finally, we show that the traveling-wave reductions lead to nonlinear maximally symmetric equations which can be linearized. The analytic solution in this case is expressed in closed-form algebraic form. Full article

The present research offers reliable analytical solutions for time-fractional linear and nonlinear dispersive Korteweg–de Vries (dKdV)-type equations by employing the Natural Generalized Laplace Transform Decomposition Method (NGLTDM). The nonlinear differential dispersive Korteweg–de Vries (dKdV) equation involves a nonlinear derivative term that depends on $\varphi$ and its partial derivative with respect to x. We employ Adomian polynomials to deal with this nonlinear part, and we utilize the Caputo derivative to illustrate the fractional part of the equation. The work provides exact theorems regarding the stability, convergence, and accuracy of the generated solutions. Illustrative examples demonstrate the effectiveness and precision of the method by delivering solutions for quickly converging series with easily calculable coefficients. We use Maple 2021 software to show graphical comparisons between the approximate and exact solutions to show how rapidly the method converges. Full article

We present a hybrid parallel scheme for efficiently solving Caputo time-fractional partial differential equations (CTFPDEs) with integer-order spatial derivatives on multicore CPU and GPU platforms. The approach combines a second-order spatial discretization with the $L1$ time-stepping scheme and employs MATLAB parfor parallelization to achieve significant reductions in runtime and memory usage. A theoretical third-order convergence rate is established under smooth-solution assumptions, and the analysis also accounts for the loss of accuracy near the initial time $t=t_0$ caused by weak singularities inherent in time-fractional models. Unlike many existing approaches that rely on locally convergent strategies, the proposed method ensures global convergence even for distant or randomly chosen initial guesses. Benchmark problems from fractional biological models—including glucose–insulin regulation, tumor growth under chemotherapy, and drug diffusion in tissue—are used to validate the robustness and reliability of the scheme. Numerical experiments confirm near-linear speedup on up to four CPU cores and show that the method outperforms conventional techniques in terms of convergence rate, residual error, iteration count, and efficiency. These results demonstrate the method’s suitability for large-scale CTFPDE simulations in scientific and engineering applications. Full article

This work analyzes the aero-elastic oscillations of cantilever beams reinforced by carbon nanotubes (CNTs). Four different distributions of single-walled CNTs are assumed as the reinforcing phase, in the thickness direction of the polymeric matrix. A modified third-order piston theory is used as an accurate tool to model the supersonic air flow, rather than a first-order piston theory. The nonlinear dynamic equation governing the problem accounts for Von Kármán-type nonlinearities, and it is derived from Hamilton’s principle. Then, the Galerkin decomposition technique is adopted to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation. This is solved analytically according to a multiple time scale method. A comprehensive parametric analysis was conducted to assess the influence of CNT volume fraction, beam slenderness, Mach number, and thickness ratio on the fundamental frequency and lateral dynamic deflection. Results indicate that FG-X reinforcement yields the highest frequency response and lateral deflection, followed by UD and FG-A patterns, whereas FG-O consistently exhibits the lowest performance metrics. An increase in CNT volume fraction and a reduction in slenderness ratio enhance the system’s stiffness and frequency response up to a critical threshold, beyond which a damped beating phenomenon emerges. Moreover, higher Mach numbers and greater thickness ratios significantly amplify both frequency response and lateral deflections, although damping rates tend to decrease. These findings provide valuable insights into the optimization of CNTR composite structures for advanced aeroelastic applications under supersonic conditions, as useful for many engineering applications. Full article

The Navier equations are reformulated to be third-order partial differential equations. New anti-Cauchy-Riemann equations can express a general solution in 2D space for incompressible materials. Based on the third-order solutions in 3D space and the Boussinesq–Galerkin method, a third-order method of fundamental solutions (MFS) is developed. For the 3D Navier equation in linear elasticity, we present three new general solutions, which have appeared in the literature for the first time, to signify the theoretical contributions of the present paper. The first one is in terms of a biharmonic function and a harmonic function. The completeness of the proposed general solution is proven by using the solvability conditions of the equations obtained by equating the proposed general solution to the Boussinesq–Galerkin solution. The second general solution is expressed in terms of a harmonic vector, which is simpler than the Slobodianskii general solution, and the traditional MFS. The main achievement is that the general solution is complete, and the number of harmonic functions, three, is minimal. The third general solution is presented by a harmonic vector and a biharmonic vector, which are subjected to a constraint equation. We derive a specific solution by setting the two vectors in the third general solution as the vectorizations of a single harmonic potential. Hence, we have a simple approach to the Slobodianskii general solution. The applications of the new solutions are demonstrated. Owing to the minimality of the harmonic functions, the resulting bases generated from the new general solution are complete and linearly independent. Numerical instability can be avoided by using the new bases. To explore the efficiency and accuracy of the proposed MFS variant methods, some examples are tested. Full article

A numerical method using advanced nonlinear shear is used to study the thermal vibration of functionally graded material (FGM) thick circular cylindrical shells. The third-order shear deformation theory (TSDT) of displacements is applied and the equations are derived of the motion of cylindrical shells and the expression of the advanced nonlinear varied shear factor. The expressions of stiffness of thick composited two-layer FGM circular cylindrical shells with sinusoidal rising temperature are applied. The partial differential equation (PDE) in dynamic equilibrium of thick FGM circular cylindrical shells is derived with respect to shear rotations and displacements under terms of thermal–mechanical loads and density inertia terms. Important parametric effects of the advanced nonlinear varied shear factor, power law index, and temperature on the stress and displacement of thick FGM circular cylindrical shells are studied. Additionally, the advanced nonlinear varied shear factor effect is included and studied for a vibrating frequency using a fully homogeneous equation. Full article

This study investigates the three-dimensional, steady, laminar boundary-layer equations of a non-Newtonian fluid over a flat plate in the absence of body forces. The classical boundary-layer theory, introduced by Prandtl in 1904, suggests that fluid flows past a solid surface can be divided into two regions: a thin boundary layer near the surface, where steep velocity gradients and significant frictional effects dominate, and the outer region, where friction is negligible. Within the boundary layer, the velocity increases sharply from zero at the surface to the freestream value at the outer edge. The boundary-layer approximation significantly simplifies the Navier–Stokes equations within the boundary layer, while outside this layer, the flow is considered inviscid, resulting in even simpler equations. The viscoelastic properties of the fluid are modeled using the Rivlin–Ericksen tensors. Lie group analysis is applied to reduce the resulting third-order nonlinear system of partial differential equations to a system of ordinary differential equations. Finally, we determine the admissible forms of the freestream velocities in the x- and z-directions. Full article

A numerical, generalized differential quadrature (GDQ) method is presented on applied heat vibration for a thick-thickness magnetostrictive functionally graded material (FGM) plate coupled with a cylindrical shell. A nonlinear c₁ term in the z axis direction of a third-order shear deformation theory (TSDT) displacement model is applied into an advanced shear factor and equation of motions, respectively. The equilibrium partial differential equation used for the thick-thickness magnetostrictive FGM layer plate coupled with the cylindrical shell under thermal and magnetostrictive loads can be implemented into the dynamic GDQ discrete equations. Parametric effects including nonlinear term coefficient of TSDT displacement field, advanced nonlinear varied shear coefficient, environment temperature, index of FGM power law and control gain on displacement, and stress of the thick magnetostrictive FGM plate coupled with cylindrical shell are studied. The vibrations of displacement and stress can be controlled by the control gain algorithms in velocity feedback control law. Full article

This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling complicated equations. To verify the applicability and validity of this method, some ideal problems of dispersive KDV-type equations are discussed and the outcoming approximate solutions are stated in sequential form. The results show that the MGLTDM is a dependable and powerful technique to deal with physical problems in diverse implementations. Full article

When studying the boundary value problems’ solvability for some partial differential equations encountered in applied mathematics, we frequently need to create systems of partial differential equations and explicitly construct linearly independent solutions explicitly for these systems. Hypergeometric functions frequently serve as solutions that satisfy these systems. In this study, we develop self-similar solutions for a third-order multidimensional degenerate partial differential equation. These solutions are represented using a generalized confluent Kampé de Fériet hypergeometric function of the third order. Full article

In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order $\mathsf{\beta }$ is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier–Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of $\mathsf{\beta }$ and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order $\mathsf{\beta }$ becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution $\mathrm{a}\mathrm{n}\mathrm{d}$ observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation. Full article

In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time. Full article