Search Results (62)

This study develops an analytical framework for investigating in-plane nonlinear subharmonic resonance in fixed–fixed circular arches under a vertical base-excitation, a phenomenon not adequately addressed in previous research. Based on Hamilton’s principle, the governing partial differential equation for in-plane nonlinear motion is first derived. The tangential displacement is then expressed as a modal superposition, and the system is reduced to a set of second-order ordinary differential equations via the Galerkin method. Using the method of multiple scales, the nonlinear 1/2-subharmonic resonance is solved, yielding closed-form, steady-state amplitude–phase relations and corresponding stability conditions. Validation against finite element simulations and Runge–Kutta analyses confirms the accuracy of the proposed approach. Dimensionless fundamental frequencies match finite element results exactly, with discrepancies in critical base-excitation below 2.5%. A close agreement is observed in both the amplitude–frequency and force–response curves with numerical predictions and Bolotin’s method, accurately capturing the characteristic hardening nonlinearity and three distinct dynamic regions spanning negligible vibration, stable resonance, and instability. Parametric studies further reveal key trends. Larger included angles intensify the vibration amplitude and promote saddle-node bifurcation, while narrowing stable operating regions. Higher slenderness ratios enhance structural flexibility and nonlinearity, shifting resonant peaks toward higher frequencies. Increased damping suppresses the response amplitude and raises the thresholds for vibration initiation and bifurcation. Full article

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-momentum relation to determine new families of nonlinear partial differential relations. The Planck–de Broglie duality principle arises from Planck’s energy expression $e=h\nu$, de Broglie’s equation for momentum $p=h/\lambda$, and Einstein’s special relativity energy, where h is the Planck constant, $\nu$ and $\lambda$ are the frequency and wavelength, respectively, of an associated wave having a wave speed $w=\nu \lambda$. The author has extended these relations to a family that is characterised by a second fundamental constant $h^{\prime }$ and underpinned by Lorentz invariant power-law particle energy-momentum expressions. In this note, we apply generalized Schrödinger operator relations and the power-law relations to generate a new family of nonlinear partial differential equations that are characterised by the constant $\kappa =h^{\prime }/h$ such that $\kappa =0$ corresponds to the Klein–Gordon equation. The resulting partial differential equation is unusual in the sense that it admits a stretching symmetry giving rise to both similarity solutions and simple harmonic travelling waves. Three simple solutions of the partial differential equation are examined including a separable solution, a travelling wave solution, and a similarity solution. A special case of the similarity solution admits zeroth-order Bessel functions as solutions while generally, it reduces to solving a nonlinear first-order ordinary differential equation. Full article

It is well known that nonlinear partial differential equations (NLPDEs) can only be solved numerically and that fourth-order NLPDEs in their derivatives require unconventional methods. This paper explains spectral numerical methods for obtaining a numerical solution by Fast Fourier Transform (FFT), implemented under Python in tis version 3.1 and their libraries (NumPy, Tkinter). Examples of NLPDEs typical of Condensed Matter Physics to be solved numerically are the conserved Cahn–Hilliard, Swift–Hohenberg and conserved Swift–Hohenberg equations. The last two equations are solved by the first- and second-order exponential integrator method, while the first of these equations is solved by the conventional FFT method. The Cahn–Hilliard equation, a phase-field model with an extended Ginzburg–Landau-like functional, is solved in two-dimensional (2D) to reproduce the evolution of the microstructure of an amorphous alloy Ce₇₅Al_{25 − x}Gax, which is compared with the experimental micrography of the literature. Finally, three-dimensional (3D) simulations were performed using numerical solutions by FFT. The second-order exponential integrator method algorithm for the Swift–Hohenberg equation implementation is successfully obtained under Python by FFT to simulate different 3D patterns that cannot be obtained with the conventional FFT method. All these 2D/3D simulations have applications in Materials Science and Engineering. Full article

In this research paper, the negative order Korteweg–de Vries equation expressed as nonlinear partial differential equation, firstly introduced by Wazwaz, is solved for the exact Jacobi elliptic function solution. For this purpose, the Jacobi elliptic function scheme, one of the direct algebraic methods, was used. The obtained exact solutions of the negative-order Korteweg–de Vries equation, a symmetry evolution equation, contains the combination of Jacobi elliptic functions and incomplete elliptic integral of second function. The three unique families of exact solutions are classified and presented. The degeneration of the obtained Jacobi elliptic function solutions into various solitons, periodic and rational solutions, is reported using the modulus transformation of Jacobi elliptic function solutions. The necessary condition existence of certain Jacobi elliptic function solutions is presented. The two-dimensional graphs for certain Jacobi elliptic function solutions are drawn to show the variation in wave propogation with respect to velocity and modulus. The non-existence of certain Jacobi elliptic function solutions for negative-order Korteweg–de Vries equations is reported. Finally, the obtained solutions were compared with the previously obtained solutions of negative-order Korteweg–de Vries equation. Full article

This paper presents a computational framework for resolving a nonlinear extension of the Black–Scholes partial differential equation that accounts for transaction costs through a volatility function dependent on the Gamma of the option price. A meshfree radial basis function-generated finite difference procedure is developed using a modified multiquadric kernel. Analytical weight formulas for first- and second-order differentiations are discussed on 3-node stencils for both uniform and non-uniform point distributions. The proposed method offers an efficient scheme suitable for accurately pricing European scenarios when nonlinear transaction cost effects. Full article

This study investigates the potential Kadomtsev–Petviashvili equation incorporating a power-type nonlinearity (PKPp), a model that features prominently in various nonlinear phenomena encountered in physics and applied mathematics. A complete Noether symmetry classification of the PKPp equation is conducted, revealing four distinct scenarios based on different values of the exponent $p$, namely, the general case where $p\ne 1,-1,-2,$ and three special cases where $p=1,$$p=-1,$ and $p=-2.$ Corresponding to each case, conservation laws are derived through a second-order Lagrangian framework. Furthermore, Lie group analysis is employed to reduce the nonlinear partial differential Equation (NLPDE) to ordinary differential Equations (ODEs), thereby enabling the effective application of the Kudryashov method and direct integration techniques to construct exact solutions. In particular, exact solutions of of the considered nonlinear partial differential equation are obtained for the cases $p=1$ and $p=2,$ illustrating the practical implementation of the proposed approach. The solutions obtained include solitary wave, periodic, and rational-type solutions. These results enhance the analytical understanding of the PKPp equation and contribute to the broader theory of nonlinear dispersive equations. Full article

A mesoscopic lattice Boltzmann method based on the BGK model is proposed to solve a class of two-dimensional second-order nonlinear partial differential equations by incorporating an amending function. The model provides an efficient and stable framework for simulating initial value problems of second-order nonlinear partial differential equations and is adaptable to various nonlinear systems, including strongly nonlinear cases. The numerical characteristics and evolution patterns of these nonlinear equations are systematically investigated. A D2Q4 lattice model is employed, and the kinetic moment constraints for both local equilibrium and correction distribution functions are derived in the four velocity directions. Explicit analytical expressions for these distribution functions are presented. The model is verified to recover the target macroscopic equations in the continuous limit via Chapman–Enskog analysis. Numerical experiments using exact solutions are performed to assess the model’s accuracy and stability. The results show excellent agreement with exact solutions and demonstrate the model’s robustness in capturing nonlinear dynamics. Full article

In this paper, a novel approximate triangular fuzzy finite element method (FEM) is proposed to solve the one-dimensional second-order unsteady nonlinear fuzzy partial differential Boussinesq equation. The physical problem concerns the case of the drought flow of a horizontal unconfined aquifer with a limited breath B and special boundary conditions: (a) at x = 0, the water level is equal to zero, and (b) at x = B, the flow rate is equal to zero due to the presence of an impermeable wall. The initial water table is assumed to be curvilinear, following the form of an inverse incomplete beta function. To account for uncertainties in the system, the hydraulic parameters—hydraulic conductivity (K) and porosity (S)—are treated as fuzzy variables, considering sources of imprecision such as measurement errors and human-induced uncertainties. The performance of the proposed fuzzy FEM scheme is compared with the previously developed orthogonal fuzzy FEM solution as well as with an analytical solution. The results are in close agreement with those of the other methods, with the mean error of the analytical solution found to be equal to $1.19·{10}^{-6}$. Furthermore, the possibility theory is applied and fuzzy estimators constructed, leading to strong probabilistic interpretations. These findings provide valuable insights into the hydraulic properties of unconfined aquifers, aiding engineers and water resource managers in making informed and efficient decisions for sustainable hydrological and environmental planning. Full article

This study presents a novel fifth-order iterative method for solving nonlinear systems derived from a modified combination of Jarratt and Newton schemes, incorporating a frozen derivative of the Jacobian. The method is applied to approximate solutions of the nonlinear convection–diffusion equation. A MATLAB script function was developed to implement the approach in two stages: first, discretizing the equation using the Crank–Nicolson Method, and second, solving the resulting nonlinear systems using Newton’s iterative method enhanced by a three-step Jarratt variant. A comprehensive analysis of the results highlights the method’s convergence and accuracy, comparing the numerical solution with the exact solution derived from linear parabolic partial differential transformations. This innovative fifth-order method provides an efficient numerical solution to the nonlinear convection–diffusion equation, addressing the problem through a systematic methodology that combines discretization and nonlinear equation solving. The study underscores the importance of advanced numerical techniques in tackling complex problems in physics and mathematics. Full article

Flow separation is a fundamental phenomenon in fluid mechanics governed by the Navier–Stokes equations, which are second-order partial differential equations (PDEs). This phenomenon significantly impacts aerodynamic performance in various applications across the aerospace sector, including micro air vehicles (MAVs), advanced air mobility, and the wind energy industry. Its complexity arises from its nonlinear, multidimensional nature, and is further influenced by operational and geometrical parameters beyond Reynolds number (Re), making accurate prediction a persistent challenge. Traditional models often struggle to capture the intricacies of separated flows, requiring advanced simulation and prediction techniques. This review provides a comprehensive overview of strategies for enhancing aerodynamic design by improving the understanding and prediction of flow separation. It highlights recent advancements in simulation and machine learning (ML) methods, which utilize flow field databases and data assimilation techniques. Future directions, including physics-informed neural networks (PINNs) and hybrid frameworks, are also discussed to improve flow separation prediction and control further. Full article

This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude Q multiplied by $e^{i\Phi }$, $\Phi$ being linear. Then, Q is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. In the other five cases, the solution is a traveling wave. It satisfies Abel-type ODE of the second kind, the first order ODE of the elliptic functions (the Weierstrass or Jacobi functions), the Airy equation, the Emden–Fawler equation, etc. At the end of the paper a short survey on the Jacobi elliptic and Weierstrass functions is included. Full article

In this paper, we show the maximal regularity of nonlinear second-order hyperbolic boundary differential equations. We aim to show if the given second-order partial differential operator satisfies the specific ellipticity condition; additionally, if solutions of the function, which are related to the first-order time derivative, possess no poles nor algebraic branch points, then the maximal regularity of nonlinear second-order hyperbolic boundary differential equations exists. This study explores the use of taking the positive definite second-order operator as the generator of an analytic semi-group. We impose specific boundary conditions to make this positive definite second-order operator self-adjoint. As a linear operator, the self-adjoint operator satisfies the linearity property. This, in turn, facilitates the application of semi-group theory and linear operator theory. Full article

This work explores the numerical translation of the weak or integral solution of nonlinear partial differential equations into a numerically efficient, time-evolving scheme. Specifically, we focus on partial differential equations separable into a quasilinear term and a nonlinear one, with the former defining the Green function of the problem. Utilizing the Green function under a short-time approximation, it becomes possible to derive the integral solution of the problem by breaking it into three integral terms: the propagation of initial conditions and the contributions of the nonlinear and boundary terms. Accordingly, we follow this division to describe and separately analyze the resulting algorithm. To ensure low interpolation error and accurate numerical Green functions, we adapt a piecewise interpolation collocation method to the integral scheme, optimizing the positioning of grid points near the boundary region. At the same time, we employ a second-order quadrature method in time to efficiently implement the nonlinear terms. Validation of both adapted methodologies is conducted by applying them to problems with known analytical solution, as well as to more challenging, norm-preserving problems such as the Burgers equation and the soliton solution of the nonlinear Schrödinger equation. Finally, the boundary term is derived and validated using a series of test cases that cover the range of possible scenarios for boundary problems within the introduced methodology. 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

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented. Full article