Symbolic Computation System Research Articles

Interaction solutions of (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation via bilinear method

Using the bilinear neural network method (BNNM) and the symbolic computation system Mathematica, this paper explains how to find an exact solution for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. In terms of activation function and weight coefficient, BNNM is a more appealing option for users than traditional symbolic computation methods. It is possible to develop a wide range of solutions and expand the classes of exact solutions by modifying the activation function. The activation function’s versatility allows it to generate a wide range of solutions with several theoretical and practical uses. The analytical solution is obtained by using a double layer type, while the rogue wave solution and mixed solutions are obtained by using a single layer type. The evolution of these waves is then illustrated using various 3D graphs, 2D graphs, and density plots.

(G'/G)-EXPANSION METHOD TO SEEK TRAVELING WAVE SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PDES ARISING IN NATURAL SCIENCES

The (G'/G)-expansion method with the aid of symbolic computational system can be used to obtain the traveling wave solutions (hyperbolic, trigonometric and rational solutions) for nonlinear time-fractional evolution equations arising in mathematical physics and biology. In this work, we will process the analytical solutions of the time-fractional classical Boussinesq equation, the time-fractional Murray equation, and the space-time fractional Phi-four equation. With the fact that the method which we will propose in this paper is also a standard, direct and computerized method, the exact solutions for these equations are obtained.

New modified tanh-function method for nonlinear evolution equations

The aim of this paper is to calculate the Traveling waves solutions by using a new technique which is called modified tanh-function method, which are successfully performed to get analytical solutions for Korteweg-deVries (KdV)–Burgers’ equation and (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. As a result, when the equation parameters are taken as special values, some new solitary wave solution are obtained. Moreover we find in this work that the modified tanh-function method give some new results which are easier and faster to compute by the help of a symbolic computation system. The results obtained were compared with standard tanh-function method.

Exact Solution of ( 4 + 1 )-Dimensional Boiti–Leon–Manna–Pempinelli Equation

Based on the Hirota bilinear method, using the heuristic function method and mathematical symbolic computation system, various exact solutions of the ( 4 + 1 )-dimensional Boiti–Leon–Manna–Pempinelli equation including the block kink wave solution, block soliton solution, periodic block solution, and new composite solution are obtained. Upon selection of the appropriate parameters, three-dimensional and contour diagrams of the exact solution were generated to illustrate their properties.

Airy function solution of two classes of nonlinear evolution equations

In this paper , we study the problem of constructing the new solutions of the mKdV equation and STO equation. By using the Airy equation and its solution, the new solutions of mKdV equation and STO equation are constructed. First of all, through function transformation, mKdV equation and STO equation can be turned into Airy function to solve. Then, construct the solution of mKdV equation and STO equation by using the solution of Airy function. Finally, analyze the properties of these solutions with the help of symbolic computation system: Mathematica.

Lump solutions to a generalized nonlinear PDE with four fourth-order terms

Abstract A combined fourth-order (2 + 1)-dimensional nonlinear partial differential equation which contains four fourth-order nonlinear terms and all second-order linear terms is formulated. This equation covers three generalized KP, Hirota–Satsuma–Ito, and Calogero–Bogoyavlenskii–Schiff equations as examples, which have physical applications in the study of various nonlinear phenomena in nature. In terms of some settings of the coefficients, a class of lump solutions is constructed by the Hirota bilinear method and the solutions are calculated through the symbolic computation system of Maple. Meanwhile, the relation between the coefficients and the solution is explored. Two special lump solutions are generated by taking proper values for the involved coefficients and parameters, and their dynamic behaviors are studied, as illustrative examples. The primary advantage of the Hirota bilinear method is to transform a nonlinear equation into a bilinear one so that the targeted equation can be easily studied.

Superposition solutions to a (3+1)-dimensional variable-coefficient Sharma-Tasso-Olver-Like equation

In this paper, the superposition solutions of (3+1)-dimensional variable-coefficient Sharma-Tasso-Olver-Like(vcSTOL) equation are studied. The equation can illustrate various difficult sciences areas. Due to the wide application, it is very important to find the exact solutions of it. By introducing transformation, the equation is transformed into bilinear form. We use variable separation method and trial function method to obtain the superposition solutions of the equation containing different functions and forms The images are drawn with the help of symbolic computing system Mathematica, and the properties of the solutions are analyzed. The analysis shows that different functions will affect the overall shape of waves, including the interaction between waves, the size, the direction and the number of waves, which can get more new phenomena. To our knowledge, those types of superposition solutions of (3+1)-dimensional vcSTOL equation mentioned in our work by variable separation method have not been reported before. Furthermore, we add the square terms to the expansion function, so that the obtained solutions have the characteristics of Lump solution, which has not been done in the previous literatures.

Автоматизированная генерация каскадных моделей турбулентности методами компьютерной алгебры

Описана разработанная методика генерации уравнений каскадных моделей турбулентности с помощью систем компьютерной алгебры. Методика позволяет варьировать размер масштабной нелокальности модели, вид квадратичных законов сохранения и спектральных законов, знаменатель геометрической прогрессии масштабов. Ее использование позволяет быстро и безошибочно генерировать целые классы моделей. Может использоваться для разработки каскадных моделей гидродинамических, магнитогидродинамических и конвективных турбулентных систем. There is a great variety of shell turbulence models. Such models reproduce certain characteristics of turbulence. A model that could reproduce all turbulence regimes does not exist at the moment. Information about a particular model is contained in a set of persistent quantities, which are some quadratic forms of turbulent fields. These quadratic forms should be formal analogs of the exact conserved quantities. It is important to note that the main idea of Shell models presupposes a refusal to describe the geometric structure of movements. At the same time, it is well known that turbulent processes in spaces of two and three dimensions behave differently. Therefore, the provision of certain combinations of conserved quantities allows indirect introducing into the shell model the information about the dimension of the physical space in which the turbulent process develops. Purpose. The aim of this work was to create software tools that would quickly generate classes of models that satisfy one or another set of conservation laws. The choice of a specific model within these classes can then be specified using additional physical considerations, for example, the existence of a given probability distribution for the interaction of certain shells. Methods. The developed technique for generating equations of shell turbulence models is carried out using symbolic computation systems (computer algebra systems - CAS). Note that symbolic packages are used not for studying ready-made shell models, but for the automated generation of the equations of these models themselves. The technique allows varying the value of the scale nonlocality of the model, the form of the quadratic conservation laws and spectral laws, the denominator of the geometric progression of scales. It allows quickly and accurately generating the entire set of classes of the models. It can be used to develop shell models of hydrodynamic, magnetohydrodynamic and convective turbulent systems. Findings. It seems that the proposed technique will be useful for studying the properties of turbulence in the framework of cascade models

A study of Bogoyavlenskii’s (2+1)-dimensional breaking soliton equation: Lie symmetry, dynamical behaviors and closed-form solutions

This paper employs the Lie symmetry analysis to investigate novel closed-form solutions to a (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation. This Lie symmetry technique, used in combination with Maple’s symbolic computation system, demonstrates that the Lie infinitesimals are dependent on five arbitrary parameters and two independent arbitrary functions f1(t) and f2(t). The invariance criteria of Lie group analysis are used to construct all infinitesimal vectors, commutative relations of their examined vectors, a one-dimensional optimal system and then several symmetry reductions. Subsequently, Bogoyavlenskii’s breaking soliton (BBS) equation is reduced into several nonlinear ODEs by employing desirable Lie symmetry reductions through optimal system. Explicit exact solutions in terms of arbitrary independent functions and other constants are obtained as a result of solving the nonlinear ODEs. These established results are entirely new and dissimilar from the previous findings in the literature. The physical behaviors of the gained solutions illustrate the dynamical wave structures of multiple solitons, curved-shaped wave–wave interaction profiles, oscillating periodic solitary waves, doubly-solitons, kink-type waves, W-shaped solitons, and novel solitary waves solutions through 3D plots by selecting the suitable values for arbitrary functional parameters and free parameters based on numerical simulation. Eventually, the derived results verify the efficiency, trustworthiness, and credibility of the considered method.

Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation

A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

Physics Informed by Deep Learning: Numerical Solutions of Modified Korteweg-de Vries Equation

In this paper, with the aid of symbolic computation system Python and based on the deep neural network (DNN), automatic differentiation (AD), and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization algorithms, we discussed the modified Korteweg-de Vries (mkdv) equation to obtain numerical solutions. From the predicted solution and the expected solution, the resulting prediction error reaches10−6. The method that we used in this paper had demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations and also opens the way for us to understand more physical phenomena later.

Traveling wave solutions for the fractional Wazwaz–Benjamin–Bona–Mahony model in arising shallow water waves

In this manuscript, we employed an extended modified auxiliary equation mapping (EMAEM) method to the 3D fractional Wazwaz–Benjamin–Bona–Mahony equation (WBBM) with the help of computer symbolic computing system. By using this technique, we get new sets of solutions like kink and anti kink, periodic and doubly periodic, bell shaped, trigonometric functional solutions, hyperbolic solutions, singular kink, rational solutions, and combined soliton like solutions. These results are figured out graphically by using suitable values of parameters with detailed behavior of physical structure of solutions.

Variational formulation and differential quadrature finite element for freely vibrating strain gradient Kirchhoff plates

AbstractIn this paper, we apply the energy variational principle to arrive at the differential equation of motion and all appropriate boundary conditions for strain gradient Kirchhoff micro‐plates. The resulting sixth‐order boundary value problem of free vibration is solved by a thirty‐six‐DOF four‐node differential quadrature plate finite element. The C2‐continuity condition of the deflection is guaranteed by devising a sixth‐order differential quadrature‐ based geometric mapping scheme that can transform the displacement parameters at Gauss‐Lobatto quadrature points into those at element nodes. The total potential energy of a generic micro‐plate element is firstly discretized in terms of nodal parameters. It is then minimized to obtain the formulation of element stiffness and mass matrices. For comparison reasons, a Hermite interpolation‐based strain gradient finite element is provided. With the help of the symbolic computation system Maple, the explicit algebraic relationship between the stiffness (or mass) matrices of two types of elements is derived. Convergence and comparison studies are conducted to show the efficacy of our element in the free vibration analysis of macro/micro‐ plates. Finally, we apply the developed method to study the size‐dependent vibration behavior of micro‐plates with uniform or stepped thickness. Numerical examples reveal that strain gradient effects can change the vibration mode shapes, not the vibration frequencies alone.

Soliton solutions of Kundu-Eckhaus equation in birefringent optical fiber with inter-modal dispersion

We study the dynamics of optical soliton in birefringent optical fiber with the effect of inter-modal dispersion. We consider the model of the Kundu-Eckhaus (KE) equation for birefringent optical fiber with the inclusion of inter-modal dispersion. Multisoliton solutions of KE equation with the presence of inter-modal dispersion are found analytically by using mapping method and a symbolic computation system. Initially, the solutions are expressed in terms of the Jacobi elliptic function which is also recovered by the use of limiting case of modulus of ellipticity. With the help of graphical illustrations of our solutions, we understand the effect of inter-modal dispersion on the soliton evolution in optical birefringent optical fiber.

Study of lump solutions to an extended Calogero-Bogoyavlenskii-Schiff equation involving three fourth-order terms

A new generalized fourth-order nonlinear differential equation originating from the Calogero-Bogoyavlenskii-Schiff equation with an extra term is studied. In terms of the coefficients of this combined nonlinear equation, a class of lump solutions is constructed by the Hirota bilinear method and calculated through the symbolic computation system of Maple. Furthermore, the affection of the extended item on the solution is explored. Two particular lump solutions with special choices of the involved parameters are generated and plotted, as illustrative examples.

New types of interaction solutions to the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation

In this paper, with the aid of symbolic computation system Maple, and based on the simplified Hirota method and ansatz technique, we discussed the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation with [Formula: see text] to obtain lump solutions, lump–kink solutions and three classes of interaction solutions. Comparing our new results with other researchers’ results shows that using this method gives the more opportunity to solve the nonlinear partial differential equations that appear in mathematics, physics, biological engineering and other fields. We also presented profiles of new lump solution, lump–kink solutions and interaction solutions as illustrative examples.

Dark optical solitons to the Biswas–Arshed equation with high order dispersions and absence of the self-phase modulation

A new model referred to as the Biswas–Arshed (BA) equation with high order dispersions for soliton transmission through optical fibers in which the self-phase modulation is fully absent is investigated, in the current article. In this respect, by using an ansatz in the complex form and distinguishing the real and imaginary parts, the resulting nonlinear ODE in the real domain is solved by the expa-function method and the symbolic computation system. As a consequence, a number of optical dark soliton solutions along with their dynamical behaviors to the BA equation are formally constructed.

Exact solitary wave solutions to the (2 + 1)-dimensional generalised Camassa–Holm–Kadomtsev–Petviashvili equation

In this paper, a ( $$2{+}1$$ )-dimensional nonlinear evolution equation (NLEE), namely the generalised Camassa–Holm–Kadomtsev–Petviashvili equation (gCHKP) or Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation (KP-BBM), is examined. After applying the newly developed generalised exponential rational function method (GERFM), 14 travelling wave solutions are formally generated. It is worth mentioning that by specifying values to free parameters some previously obtained solutions can be recovered. The simplest equation method (SEM) is used to prove that the solutions obtained by GERFM are good. With the aid of a symbolic computation system, we prove that GERFM is more efficient and faster.

New extended direct algebraic method for the Tzitzica type evolution equations arising in nonlinear optics

In this study, the new extended direct algebraic method is exerted for constructing more general exact solutions of the three nonlinear evolution equations with physical interest namely, the Tzitzeica equation, the Dodd-Bullough-Mikhailor equation and the Liouville equation. By using of an appropriate traveling wave transformation reduces these equations to ODE. We state that this method is excellently a generalized form to obtain solitary wave solutions of the nonlinear evolution equations that are widely used in theoretical physics. The method appears to be easier and faster by means of symbolic computation system.

COMPARATIVE STUDY OF WAVELET METHODS FOR SOLVING BERNOULLI’S EQUATIONS

A comparative study of two numerical techniques is presented for solving nonlinear differential equations of the Bernoulli’s type. Proposed techniques are based on the conversion of nonlinear differential equations into linear differential equations by substituting particular factor and utilization of Haar wavelet collocation method (HWCM) and Hermite wavelet collocation method (HeWCM) to these linear equations. Searching for numerical solutions of such equations has attracted a considerable amount of research work where computer symbolic systems facilitate the computational work.