Showing papers on "Partial differential equation published in 2018"
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TL;DR: A deep learning-based approach that can handle general high-dimensional parabolic PDEs using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function.
Abstract: Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black–Scholes equation, the Hamilton–Jacobi–Bellman equation, and the Allen–Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.
1,309 citations
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TL;DR: A deep learning algorithm similar in spirit to Galerkin methods, using a deep neural network instead of linear combinations of basis functions is proposed, and is implemented for American options in up to 100 dimensions.
1,290 citations
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TL;DR: In this article, a new paradigm of learning partial differential equations from small data is presented, which is essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equation, to extract patterns from high-dimensional data generated from experiments.
986 citations
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14 Feb 2018
TL;DR: Deep Ritz Method as mentioned in this paper is a deep learning-based method for numerically solving variational problems, particularly the ones that arise from partial differential equations, and has the potential to work in rather high dimensions.
Abstract: We propose a deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz Method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.
848 citations
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TL;DR: In this article, a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations are analyzed, based on the Hirota bilinear formulation and the primary object is the class of positive multivariate quadrastic functions.
543 citations
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TL;DR: This paper shows how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator.
457 citations
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TL;DR: This work puts forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time by approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks.
Abstract: We put forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time. Specifically, we approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks. The first network acts as a prior on the unknown solution and essentially enables us to avoid numerical differentiations which are inherently ill-conditioned and unstable. The second network represents the nonlinear dynamics and helps us distill the mechanisms that govern the evolution of a given spatiotemporal data-set. We test the effectiveness of our approach for several benchmark problems spanning a number of scientific domains and demonstrate how the proposed framework can help us accurately learn the underlying dynamics and forecast future states of the system. In particular, we study the Burgers', Korteweg-de Vries (KdV), Kuramoto-Sivashinsky, nonlinear Schrodinger, and Navier-Stokes equations.
395 citations
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TL;DR: The method extracts a reduced basis from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD) and employs artificial neural networks (ANNs) to accurately approximate the coefficients of the reduced model.
315 citations
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TL;DR: In this article, a new solitons solution of the complex Ginzburg-Landau equation with Kerr law nonlinearity was found using the new extended direct algebraic method.
212 citations
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TL;DR: In this paper, a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time is proposed, where the unknown solution as well as the nonlinear dynamics are approximated by two deep neural networks.
Abstract: A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical models of the physical world. In the current era of abundance of data and advanced machine learning capabilities, the natural question arises: How can we automatically uncover the underlying laws of physics from high-dimensional data generated from experiments? In this work, we put forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time. Specifically, we approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks. The first network acts as a prior on the unknown solution and essentially enables us to avoid numerical differentiations which are inherently ill-conditioned and unstable. The second network represents the nonlinear dynamics and helps us distill the mechanisms that govern the evolution of a given spatiotemporal data-set. We test the effectiveness of our approach for several benchmark problems spanning a number of scientific domains and demonstrate how the proposed framework can help us accurately learn the underlying dynamics and forecast future states of the system. In particular, we study the Burgers', Korteweg-de Vries (KdV), Kuramoto-Sivashinsky, nonlinear Schr\"{o}dinger, and Navier-Stokes equations.
195 citations
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TL;DR: In this paper, the existence of solutions for two types of high order fractional integro-differential equations is studied. But the authors focus on the CFD and DCF derivations.
Abstract: By using the fractional Caputo–Fabrizio derivative, we introduce two types new high order derivations called CFD and DCF. Also, we study the existence of solutions for two such type high order fractional integro-differential equations. We illustrate our results by providing two examples.
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TL;DR: In this paper, a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order α ∆ in(0, 1)$ in time, is presented.
Abstract: We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time It relies on three technical tools: a fractional version of the discrete Gronwall type inequality, discrete maximal regularity, and regularity theory of nonlinear equations We establish a general criterion for showing the fractional discrete Gronwall inequality and verify it for the L1 scheme and convolution quadrature generated by backward difference formulas Further, we provide a complete solution theory, eg, existence, uniqueness, and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term Together with the known results of discrete maximal regularity, we derive pointwise $L^2(\Omega)$ norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order $O(h^2)$ (up to a logarithmic factor) and $O(\tau^\alpha)$,
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TL;DR: In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes.
Abstract: In 2013, a new nonlocal symmetry reduction of the well-known AKNS (an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, and Alan C. Newell et al. (1974)) scattering problem was found. It was shown to give rise to a new nonlocal PT symmetric and integrable Hamiltonian nonlinear Schrodinger (NLS) equation. Subsequently, the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localized, time periodic one-soliton solutions was found. In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes. The direct and inverse scattering problems are analyzed. Specifically, the direct problem is formulated, the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained. The inverse scattering problem, which...
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TL;DR: In this paper, the authors proposed a method to obtain exact solutions of nonlinear partial differential equations by generalizing the exponential rational function method, which can be used to obtain the exact solution of the Schrodinger equation in a relatively easy way.
Abstract: The present paper suggests a novel technique to acquire exact solutions of nonlinear partial differential equations. The main idea of the method is to generalize the exponential rational function method. In order to examine the ability of the method, we consider the resonant nonlinear Schrodinger equation (R-NLSE). Many variants of exact soliton solutions for the equation are derived by the proposed method. Physical interpretations of some obtained solutions is also included. One can easily conclude that the new proposed method is very efficient and finds the exact solutions of the equation in a relatively easy way.
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TL;DR: The reproducing kernel algorithm is introduced for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit.
Abstract: The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit.,The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions.,Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models.,Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers.,The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability.,Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest.,This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.
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TL;DR: The data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.
Abstract: Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier–Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.
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TL;DR: In this paper, a (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation that possesses a Hirota bilinear form is considered, and a class of explicit lump solutions is computed through conducting symbolic computations with Maple, and some plots of a specic presenteded lump solution are made to shed light on the characteristics of lumps.
Abstract: A (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation that possesses a Hirota bilinear form is considered. Starting with its Hirota bilinear form, a class of explicit lump solutions is computed through conducting symbolic computations with Maple, and a few plots of a specicpresented lump solution are made to shed light on the characteristics of lumps. The result provides a new example of (2 + 1)-dimensional nonlinear partial differential equations which possess lump solutions.
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TL;DR: This work approximate the unknown solution by a deep neural network which essentially enables the author to benefit from the merits of automatic differentiation in partial differential equations.
Abstract: Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, both in 100-dimensions.
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TL;DR: Stochastic homogenization theory allows us to better understand the convergence of the algorithm, and a stochastic control interpretation is used to prove that a modified algorithm converges faster than SGD in expectation.
Abstract: Entropy-SGD is a first-order optimization method which has been used successfully to train deep neural networks. This algorithm, which was motivated by statistical physics, is now interpreted as gradient descent on a modified loss function. The modified, or relaxed, loss function is the solution of a viscous Hamilton–Jacobi partial differential equation (PDE). Experimental results on modern, high-dimensional neural networks demonstrate that the algorithm converges faster than the benchmark stochastic gradient descent (SGD). Well-established PDE regularity results allow us to analyze the geometry of the relaxed energy landscape, confirming empirical evidence. Stochastic homogenization theory allows us to better understand the convergence of the algorithm. A stochastic control interpretation is used to prove that a modified algorithm converges faster than SGD in expectation.
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TL;DR: A modified L1 scheme for solving time fractional partial differential equations is introduced and error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases are obtained.
Abstract: We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c...
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TL;DR: It is proved, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.
Abstract: Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, such numerical simulations indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are also a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of these mathematical results prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$ and the PDE dimension $d \in \mathbb{N}$ and we thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.
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TL;DR: In this paper, the authors examined the effectiveness of an integration scheme called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs).
Abstract: This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method.
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TL;DR: In this article, an analytical method is proposed for the nonlinear primary resonance analysis of cylindrical shells made of functionally graded (FG) porous materials subjected to a uniformly distributed harmonic load including the damping effect.
Abstract: An analytical method is proposed for the nonlinear primary resonance analysis of cylindrical shells made of functionally graded (FG) porous materials subjected to a uniformly distributed harmonic load including the damping effect. The Young's modulus, shear modulus and density of porous materials are assumed to vary through the thickness direction based on the assumption of a common mechanical feature of the open-cell foam. Three types of FG porous distributions, namely symmetric porosity distribution, non-symmetric porosity stiff or soft distribution and uniform porosity distribution are considered in this paper. Theoretical formulations are derived based on Donnell shell theory (DST) and accounting for von-Karman strain-displacement relation and damping effect. The first mode of deflection function that satisfies the boundary conditions is introduced into this nonlinear governing partial differential equation and then a Galerkin-based procedure is utilized to obtain a Duffing-type nonlinear ordinary differential equation with a cubic nonlinear term. Finally, the governing equation is solved analytically by conducting the method of multiple scales (MMS) which results in frequency-response curves of FG porous cylindrical shells in the presence of damping effect. The detailed parametric studies on porosity distribution, porosity coefficient, damping ratio, amplitude and frequency of the external harmonic excitation, aspect ratio and thickness ratio, shown that the distribution type of FG porous cylindrical shells significantly affects primary resonance behavior and the response presents a hardening-type nonlinearity, which provides a useful help for the design and optimize of FG porous shell-type devices working under external harmonic excitation.
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TL;DR: In this article, a new extended direct algebraic method for solving the nonlinear conformable fractional Schrodinger-Hirota equation (FSHE) is presented.
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TL;DR: In this paper, the authors considered the unsteady peristaltic motion of a non-Newtonian nanofluid under the influence of a magnetic field and Hall currents and provided a theoretical model that estimates the impact of a wide range of parameters on the characteristics of blood-like fluid flows.
Abstract: In this study, we considered the unsteady peristaltic motion of a non-Newtonian nanofluid under the influence of a magnetic field and Hall currents. The simultaneous effects of ion slip and chemical reaction were also taken into consideration. The flow problem was suggested on the basis of the continuity, thermal energy, linear momentum, and nanoparticle concentration, which were further reduced with the help of Ohm's law. Mathematical modelling was executed using the lubrication approach. The resulting highly nonlinear partial differential equations were solved semi-analytically using the homotopy perturbation technique. The impacts of all the pertinent parameters were investigated mathematically and graphically. Numerical calculations have been used to calculate the expressions for the pressure increase and friction forces along the whole length of the channel. The results depict that for a relatively large value of the Brownian parameter, the chemical reaction has a dual behaviour on the concentration profile. Moreover, there is a critical point of the magnetic parameter at which the behaviours of the pressure increase and friction forces are reversed for progressive values of the power law index. The present investigation provides a theoretical model that estimates the impact of a wide range of parameters on the characteristics of blood-like fluid flows.
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TL;DR: A class of lump solutions is explicitly generated via symbolic computations with Maple, together with plots of a specific lump solution, which enlarges the category of nonlinear partial differential equations which possess lump solutions.
Abstract: A generalized Calogero–Bogoyavlenskii–Schiff equation is considered, and based on its Hirota bilinear form, a class of lump solutions is explicitly generated via symbolic computations with Maple, together with plots of a specific lump solution. The result enlarges the category of nonlinear partial differential equations which possess lump solutions.
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TL;DR: In this article, Liouville-Caputo (LC) and Atangana-Baleanu (AB) fractional operators were used for solving time-fractional partial differential equations.
Abstract: In this paper, time-fractional partial differential equations (FPDEs) involving singular and non-singular kernel are considered. We have obtained the approximate analytical solution for linear and nonlinear FPDEs using the Laplace perturbation method (LPM) defined with the Liouville-Caputo (LC) and Atangana-Baleanu (AB) fractional operators. The AB fractional derivative is defined with the Mittag-Leffler function and has all the properties of a classical fractional derivative. In addition, the AB operator is crucial when utilizing the Laplace transform (LT) to get solutions of some illustrative problems with initial condition. We show that the mentioned method is a rather effective and powerful technique for solving FPDEs. Besides, we show the solution graphs for different values of fractional order $\alpha$
, distance term x and time value t . The classical integer-order features are fully recovered if $\alpha$
is equal to 1.
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TL;DR: In this paper, a modified auxiliary equation method was presented to obtain solitary solutions of three fundamental models in the biological branch of science, i.e., the biological population model, the equal width model and modified equal width equation.
Abstract: In this article, we present a modified auxiliary equation method. We harness this modification in three fundamental models in the biological branch of science. These models are the biological population model, equal width model and modified equal width equation. The three models represent the population density occurring as a result of population supply, a lengthy wave propagating in the positive x-direction, and the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes, respectively. We discuss these models in nonlinear fractional partial differential equation formulas. We used the conformable derivative properties to convert them into nonlinear ordinary differential equations with integer order. After adapting, we applied our new modification to these models to obtain solitary solutions of them. We obtained many novel solutions of these models, which serve to understand more about their properties. All obtained solutions were verified by putting them back into the original equations via computer software such as Maple, Mathematica, and Matlab.
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TL;DR: In this article, the exp(−varphi(ξ)) expansion method was employed to construct the exact traveling wave solutions of equal width and modified equal width equations, and the obtained results are novel and have numerous applications in current areas of research in mathematical physics.
Abstract: The Equal-Width and Modified Equal-Width equations are used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. In this article we have employed extend simple equation method and the exp(−varphi(ξ)) expansion method to construct the exact traveling wave solutions of equal width and modified equal width equations. The obtained results are novel and have numerous applications in current areas of research in mathematical physics. It is exposed that our method, with the help of symbolic computation, provides a effective and powerful mathematical tool for solving different kind nonlinear wave problems.
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TL;DR: Pyomo.dae is an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks.
Abstract: We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org
. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differential equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.