Solution Of Partial Differential Equations Research Articles

An endeavor from the Glowinski-Le Tallec splitting for approximating the solution of Kawarada equation

This paper studies an extended application of the Glowinski-Le Tallec splitting for approximating solutions of linear and nonlinear partial differential equations. It is shown that the three-level, six-component operator decomposition, originally designed for Lagrangian optimizations, provides a stable second-order operator splitting approximation for the solutions of evolutional partial differential equations. It is also found that the Glowinski-Le Tallec formula not only provides an effective enhancement to conventional two-level, four-component ADI and LOD methods, but also introduces a flexible way for constructing multi-parameter operator splitting strategies in respective spaces where broad spectrums of mathematical models may exist for important natural phenomena and applications. The extended operator splitting is utilized for solving a singular and nonlinear Kawarada problem satisfactorily. Multiple simulation results are presented.

Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method

Abstract Finding analytical solutions for nonlinear partial differential equations is physically meaningful. The Kaup-Kupershmidt (KK) equation is studied in this article. The KK equation is of fifth order, such that several solitary solutions are obtained. In this article, however, the modified auxiliary function approach is applied to this model to find solitary solutions. These solutions are written in terms of Jacobi functions. Therefore, the obtained solutions can be implemented graphically to show different patterns for appropriate parameters.

Polynomial solutions of nonlinear partial differential equations in sobolev spaces

We give a proof about the existence and uniqueness of polynomial solutions for a nonlinear partial differential equations in Sobolev spaces defined in a smooth domain

COMPARATIVE STUDY OF THE LAPLACE-ADOMIAN METHOD AND THE VARIATIONAL ITERATIONS METHOD FOR DETERMINING THE EXACT SOLUTIONS OF SOME PARTIAL DIFFERENTIAL EQUATIONS

This work is a verification of the effectiveness of Laplace-Adomian and variational iterations methods for solving partial differential equations. The coupling of Laplace and Adomian methods has made it possible to exploit Adomian polynomials with Laplace transforms, as well as their inverse, to overcome the difficulties associated with the non-linearity of the equations. The variational iteration method, with its correction functional, facilitates the determination of the general Lagrange multiplier, which is essential for the rest of the solution. These methods have enabled us to obtain the exact solutions.

Machine learning for a class of partial differential equations with multi-delays based on numerical Gaussian processes

Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems.

Rigorous higher-order Poincaré optical vortex modes

The state of polarization of a general form of an optical vortex mode is represented by the vector ϵ^ m , which is associated with a vector light mode of order m>0. It is formed as a linear combination of two product terms involving the phase functions e±imϕ times the optical spin unit vectors σ∓. Any such state of polarization corresponds to a unique point (Θ P ,Φ P ) on the surface of the order m unit Poincaré sphere. However, albeit a key property, the general form of the vector potential in the Lorenz gauge A=ϵ^ m Ψ m , from which the fields are derived, including the longitudinal fields, has neither been considered nor has had its consequences been explored. Here, we show that the spatial dependence of Ψ m can be found by rigorously demanding that the product ϵ^ m Ψ m satisfies the vector paraxial equation. For a given order m this leads to a unique Ψ m , which has no azimuthal phase of the kind e i ℓ ϕ , and it is a solution of a scalar partial differential equation with ρ and z as the only variables. The theory is employed to evaluate the angular momentum for a general Poincaré mode of order m yielding the angular momentum for right- and left- circularly polarized, elliptically polarized, linearly polarized and radially and azimuthally polarized higher-order modes. We find that in applications involving Laguerre–Gaussian modes, only the modes of order m≥2 have non-zero angular momentum. All modes have zero angular momentum for points on the equatorial circle for which cos⁡Θ P =0.

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

Stability analysis of dual solutions for mixed convection and thermal radiation with hybrid nanofluid flow past shrinking/stretching curved surface

This study portrays the stability analysis and dual solutions of mixed convection and thermal radiation of hybrid nano-fluid flow past stretching/shrinking a curved surface in the presence of injection/suction conditions. A hybrid nano-fluid, in which water is used as the base fluid, copper and alumina are used as nano-particles, and the magnetic field is taken into account. The present study’s findings will provide fruitful implications for future research in the field of fluid dynamics. The bvp4c method using Matlab software is implemented to get the numerical solution of the nonlinear partial differential equation transformed into the ordinary differential equation. The behavior of the first and second solutions under the governing parameters on the curved surface of dimensionless velocity g′(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g'(\\xi )$$\\end{document}, shear stress profile g′′(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(\\xi )$$\\end{document}, temperature profile θ(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta (\\xi )$$\\end{document}, skin friction coefficient Cfs, and local Nusselt’s number Nus were visualized in figurative and tabular form. From this the following are investigated: as the values of ϕj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi j$$\\end{document} increase, the velocity profile for the second solution decreases, and the opposite trend is observed for the first solution. For the values of K and λ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda 1$$\\end{document}, the shear stress profile increased for the first solution, and the opposite trend was observed for the second solution, though after some interval points, the inverse of this statement was observed.For the values of S,Pr,Rd,M,andλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S,Pr,Rd,M, \\;\ ext{and}\\; \\lambda$$\\end{document}, the upwind thermal boundary layer of the first solution is larger than the second solution.For the value of M uphill, the estimation of the absolute value of λci\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ci$$\\end{document} increases for both the skin friction coefficient and the local Nusselt number. In the second solution, increasing the values of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}, Pr, S, Ec, and M has a similar effect on g′′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(0)$$\\end{document}, Cfs, -θ′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta '(0)$$\\end{document}, and Nus. In the first solution, increasing the values of Ec, S, and Pr on g′′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(0)$$\\end{document} and Cfs results in a decrease. The first solution has a positive eigenvalue, whereas the second solution has a negative eigenvalue. Agreement between the present analysis and literature is acceptable.

Теорема існування для задачі масопереносу на графі

The mass transfer process in a porous medium is described using the Richards-Klute equation. This equation describes mass flows due to the actions of gravity and capillarity and allows modeling the mass transfer process with saturation limit. The Richards-Klute equation is a nonlinear elliptic-parabolic partial differential equation, so the main methods for solving it and modeling the mass transfer process are numerical methods. The article considers a model of a system of interconnected pipes, inside which the process of mass transfer takes place. Such systems are often found in agriculture and are actively used in the construction of irrigation systems. The article proposes to model pipe system using graphs, where pipes are represented by graph edges, and connection points or free ends of system pipes are represented by graph vertices. The article contains the definitions of the Richards-Klute equation on a graph in the usual and weak forms. On the edges of the graph, one-dimensional cases of the Richards-Klute equation are considered, while on the vertices either the boundary conditions are given or the equation that models the law of mass conservation is given. The definitions of the solution and weak solution of the Richards-Klute equation on the graph are also given. Also, the theorem of the existence of a weak solution of the Richards-Klute equation on a graph is proved. To prove the theorem of the existence of a weak solution of the Richards-Klute equation on a graph, the Kirchhoff transformation is used and conditions are given that are analogous to the conditions used in the proof of the existence of a weak solution of the Richards-Klute equation in regular domains in three-dimensional space, and which are defined in the classical work [1], which is devoted to the problems of existence and uniqness of weak solutions of elliptic-parabolic partial differential equations.

New applications of the fractional derivative to extract abundant soliton solutions of the fractional order PDEs in mathematics physics

The motive of this research work is to unravel the mysteries of nature through fractional-order partial differential equations (PDEs). Here, we focus on two important fractional order nonlinear PDEs, namely the fractional order (4+1)-dimensional Fokas equation, which is used to give the model of many physical phenomena and dynamical processes, and the other one is the fractional order (2+1)-dimensional breaking soliton equation which is used to analyze the nonlinear problems like optical fiber communications, ocean engineering, etc. Recently, it has been an essential topic to extract the new soliton solutions which are used to investigate the hidden physical conditions of the nonlinear fractional PDEs. So, it is essential to solve those nonlinear fractional PDEs which have a physical impact in the fields of science and modern engineering. In our investigation, we attempt to provide nonlinear wave propagation patterns and investigate the equations, as mentioned earlier, through a computational method. A computing operating software called Mathematica has been applied to get a clear visualization of our gained outcomes, and we ascertain such types of shapes as the bell shape soliton, the anti-bell shape soliton, the singular bell shape soliton, the periodic solution, and the singular periodic solution. Our obtained results can keep an indispensable role in explaining various physical phenomena of nature shortly, and the applied method is the very cogent, efficient, and interesting to extract such types of solutions. Since we extract abundant solutions of these models, so we hope that this article is the best applications of mention method.

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. Non-local nonlinear PDE models can accurately capture these phenomena, but traditional numerical approximation methods are infeasible when the considered non-local PDE is high-dimensional. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman–Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

Moving iso-contour method for solving partial differential equations

The numerical solution of partial differential equations is often performed on a numerical grid, where the grid points are used for estimating the partial derivatives. The grid can be fully static as in Eulerian type of solution method, or the grid points can move during the solution, which is the case in Lagrangian type of method. In the current article, a numerical solution method is presented, where the grid points are located on iso-contours of the two-dimensional field. The method calculates the local movement of the iso-contours according to an evolution equation described by the PDE, and the solution proceeds by moving the grid points towards the calculated direction. Additional stability is obtained by setting the grid points to move along the iso-contour line. To exemplify the application of the method, numerical examples are calculated for the two-dimensional diffusion equation.

Hyena neural operator for partial differential equations

Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to solving partial differential equations that involves the use of neural operators. Neural operators are neural network architectures that learn mappings between function spaces and have the capability to solve partial differential equations based on data. This study utilizes a novel neural operator called Hyena, which employs a long convolutional filter that is parameterized by a multilayer perceptron. The Hyena operator is an operation that enjoys sub-quadratic complexity and enjoys a global receptive field at the meantime. This mechanism enhances the model’s comprehension of the input’s context and enables data-dependent weight for different partial differential equation instances. To measure how effective the layers are in solving partial differential equations, we conduct experiments on the diffusion–reaction equation and Navier–Stokes equation and compare it with the Fourier neural operator. Our findings indicate that the Hyena neural operator can serve as an efficient and accurate model for learning the partial differential equation solution operator. The data and code used can be found at https://github.com/Saupatil07/Hyena-Neural-Operator.

An Efficient Method for Solving Two-Dimensional Partial Differential Equations with the Deep Operator Network

Partial differential equations (PDEs) usually apply for modeling complex physical phenomena in the real world, and the corresponding solution is the key to interpreting these problems. Generally, traditional solving methods suffer from inefficiency and time consumption. At the same time, the current rise in machine learning algorithms, represented by the Deep Operator Network (DeepONet), could compensate for these shortcomings and effectively predict the solutions of PDEs by learning the operators from the data. The current deep learning-based methods focus on solving one-dimensional PDEs, but the research on higher-dimensional problems is still in development. Therefore, this paper proposes an efficient scheme to predict the solution of two-dimensional PDEs with improved DeepONet. In order to construct the data needed for training, the functions are sampled from a classical function space and produce the corresponding two-dimensional data. The difference method is used to obtain the numerical solutions of the PDEs and form a point-value data set. For training the network, the matrix representing two-dimensional functions is processed to form vectors and adapt the DeepONet model perfectly. In addition, we theoretically prove that the discrete point division of the data ensures that the model loss is guaranteed to be in a small range. This method is verified for predicting the two-dimensional Poisson equation and heat conduction equation solutions through experiments. Compared with other methods, the proposed scheme is simple and effective.

A new treatment of boundary conditions in PDE solution with Galerkin methods via Partial Integral Equation framework

We present a new mathematical framework for solution of Partial Differential Equations (PDEs), which is based on an exact transformation of the underlying PDE that removes the boundary constraints from the solution state and moves them into the dynamics of the equivalent transformed equation. The framework is based on a Partial Integral Equation (PIE) representation of a PDE or a system of PDEs, where Partial Integral Equation does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized and consistent treatment of boundary conditions in constructing spectrally convergent solution approximations to a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries, among others. The significance of this result is that a solution to almost any linear PDE in a form of a function series approximation can now be systematically constructed, irrespective of the boundary conditions. Furthermore, in many cases, the resulting ODE system for the expansion coefficients can now be integrated analytically in time, which allows us to obtain solution approximations to a broad class of unsteady PDEs with unprecedented accuracy. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. We also present comparison of the PIE methods with some classical direct PDE solution methods, further demonstrating advantages and potential limitations of the PIE approach. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.

Stability analysis and dynamics of solitary wave solutions of the (3+1)-dimensional generalized shallow water wave equation using the Ricatti equation mapping method

Our aim is to examine the dynamic characteristics of the (3+1)-dimensional generalized equation governing shallow water waves. When the horizontal extent of the fluid significantly surpasses the vertical dimension, the employment of shallow water equations becomes appropriate. By employing an inventive Ricatti equation mapping approach, we have obtained a range of solitary wave solutions in both explicit and generalized forms. Solitons are particularly useful in signal and energy transmission due to their ability to preserve their shape during propagation. By studying soliton solutions in nonlinear systems, we can gain valuable knowledge applicable to practical fields. To deepen our grasp of the equation’s physical implications, we have presented some solutions through 3D, 2D, and contour graphics. Employing a linear stability approach, we confirm the equation’s stability. The method used distinguishes out for its simplicity, dependability, and ability to generate novel solutions for nonlinear partial differential equations in the area of mathematical physics. The research findings reported here demonstrate the viability of the used method in studying nonlinear phenomena in the studied equation as well as other nonlinear problems in mathematical physics. By employing this tool, academics have the chance to deepen and increase their grasp of the complex mathematical concepts underlying real-world problems.

Exact Solutions of Nonlinear Partial Differential Equations Using the Extended Kudryashov Method and Some Properties

In this paper, we consider how to find new exact solutions for nonlinear partial differential equations using the extended Kudryashov method. This method mainly uses the Riccati equation and the Bernoulli equation where there are some underdetermined constant parameters. And we also use the concept of symmetry to study its reduction equation, Lie transformation group, self-adjointness, and conservation laws. This paper mainly studies the Boussinesq class and the shallow water wave equation in (1 + 1) dimensions and tries to find new exact solutions and symmetry properties of them.

Boundedness Analysis of the Fractional Maximal Operator in Grand Herz Space on the Hyperplane

The primary purpose of this work was to prove the boundedness of the Fractional Maximal Operator in Grand Herz Spaces on the Hyperplane. Here, We defined Grand Herz Space in a continuous Case. For Simplicity, We divided our Problem into two theorems by taking two subsets of Hyperplane( ) as ( ) and its complement . We proved the boundedness of the Fractional Maximal Operator in Grand Herz Space on these two subsets of Hyperplane. We also defined the continuous Case of Grand Herz Space. We proved some results to use in our proof. We represented other terms this paper uses, i.e. the Hyperplane and Fractional Maximal operator. Our proof method relied on one of the corollaries we gave in this paper. We proved the condition to apply that corollary, and then by referring to this, we confirmed both of our theorems. This paper is helpful in Harmonic analysis and delivers ways to analyse the solutions of partial differential equations. The Problem of our discussion provides methods to study the properties of very complex functions obtained from different problems from Physics, Engineering and other branches of science. Solutions of nonlinear Partial Differential equations often resulted in such functions which required deep analysis. Our work helps check the boundedness of such types of functions.

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker–Planck equation in high dimensions and the Kuramoto–Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training.

Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua

Non-prismatic slender continua are the prototypical models of many structural elements used in engineering applications, such as wind turbine blades and towers. Unfortunately, closed-form expressions for stresses and strains in such continua are much more difficult to find than in prismatic ones, e.g., the de Saint-Venant’s cylinder, for which some analytical solutions are known. Starting from a suitable mechanical model of a tapered slender continuum with one dimension much larger than the other tapered two, a variational principle is exploited to derive the field equations, i.e., the set of partial differential equations and boundary conditions that govern its state of stress and strain. The obtained equations can be solved in closed form only in a few cases. Paradigmatic examples in which analytical solutions are obtainable in terms of stresses, strains, or related mechanical quantities of interest in engineering applications are presented and discussed.