Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods

A nonlinear conformable time-fractional parabolic equation with exponential nonlinearity is explored, in this article. First, under the specific transformations, the time-fractional parabolic equation is changed into a nonlinear ODE of integer order, and then, the reduced equation is solved using two lately established techniques called the \({ \exp }\left( { - \varphi \left( \varepsilon \right)} \right)\)-expansion and modified Kudryashov methods. Several exact solutions in various wave forms for the nonlinear conformable time-fractional parabolic equation with exponential nonlinearity are formally constructed.     

New exact solutions of nonlinear conformable time-fractional Phi-4 equation

In this paper, new exact analytical solutions of time-fractional Phi-4 equation are developed using extended direct algebraic method by means of conformable fractional derivative. The obtained new results reveal that the proposed method is effective to studythe nonlinear dispersive equations in mathematical physics.     

Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation

In this Letter we propose a new generalization of the two-dimensional differential transform method that will extend the application of the method to a diffusion-wave equation with space- and time-fractional derivatives. The new generalization is based on generalized Taylor's formula and Caputo fractional derivative. Theorems that are never existed before are introduced with their proofs. Several illustrative examples are given to demonstrate the effectiveness of the obtained results. The results reveal that the technique introduced here is very effective and convenient for solving partial differential equations of fractional order.     

New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method

In this paper, the nonlinear Boussinesq equations with the conformable time-fractional derivative are solved analytically using the well-established modified Kudryashov method. As a consequence, a number of new exact solutions for this type of equations are formally derived. It is believed that the method is one of the most effective techniques for extracting new exact solutions of nonlinear fractional differential equations.     

Method for solving the nonlinear diffusion equation

We discuss the properties of the nonlinear diffusion equation which was presented for the approximation of high-concentration arsenic and boron profiles. Typical features of the profile are a flat region near the surface, a decreasing intermediate region and a tail region dropping sharply down compared with an error function complement. Earlier the nonlinear equation of the diffusion where the diffusion coefficient is directly proportional to the concentration of the impurities has been proposed. This proposal has been made on the theoretical assumption that diffusion is the result of Brownian movement and must occur with a finite velocity.     

Transformation methods for solving nonlinear field equations

The collection of extended canonical transformations of first-order contact manifolds is studied. This collection is shown to form a group under target-source composition and to contain the group of all first prolongations of point transformation of the underlying graph space and all isogroups of completely integrable horizontal ideals. Extended canonical transformations are compared and contrasted with Bäcklund transformations. These results are used to construct an extended Hamilton-Jacobi method for systems of nonlinear PDE. The collection of all extended canonical transformations is also shown to contain infinitely many one-parameter families of transformations, but there is no Lie group structure that contains these one-parameter families, in general. Conditions are obtained under which a one-parameter family of extended canonical transformations will map a solution of the fundamental ideal that characterizes a given system of PDE into a one-parameter family of solutions. These results are applied to the -Gordon equation _x1 = () and to the Navier-Stokes equations.     

一类非线性方程类孤波的近似解法

采用了一个简单而有效的技巧,研究了一类扰动发展方程.首先引入求解一个相应典型方程的类孤波近似解,然后利用泛函映射方法得到了原扰动发展方程的近似解,指出了近似解级数的收敛性,并用解析方法,讨论了近似解的精度.     

On spectral methods for solving nonlinear acoustics equations

Two very efficient methods for obtaining approximate solutions to nonlinear acoustics equations are discussed. I proposed these methods earlier, but they are still little known. The first method is based on expanding an unknown function into a Taylor series with respect to the coordinate (evolution variable) and on approximate summation of the terms of this series in all orders up to the infinite order. This series can be summed completely only in particular cases, e.g., for a simple wave. It has been noted that the partial summation technique is implemented more easily if all the terms of the series are represented as corresponding topological diagrams. The second method is based on introducing a “nonlinear” phase delay (proportional to the wave amplitude) for the temporal variable in linear solutions of the problem. The application technique of these methods is illustrated by obtaining approximate solutions of the Burgers equation.     

The modified simple equation method for solving some fractional-order nonlinear equations

Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinear fractional Klein–Gordon equation by using the modified simple equation method.     

Iterative methods for solving nonlinear problems of nuclear reactor criticality

The paper presents iterative methods for calculating the neutron flux distribution in nonlinear problems of nuclear reactor criticality. Algorithms for solving equations for variations in the neutron flux are considered. Convergence of the iterative processes is studied for two nonlinear problems in which macroscopic interaction cross sections are functionals of the spatial neutron distribution. In the first problem, the neutron flux distribution depends on the water coolant density, and in the second one, it depends on the fuel temperature. Simple relationships connecting the vapor content and the temperature with the neutron flux are used.     

强非线性发展方程孤波同伦解法

研究了一个强非线性发展方程. 利用同伦映射方法，首先构造了相应的同伦变换；其次选取了适当的初始近似；再用迭代方法得到了孤波的任意次精度的近似解.     

Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg–de Vries equation

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries(LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation, and second, we find the related exact solutions for the derived generators. Finally,according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field Kd V equation.     

求解非线性薛定谔方程的简便方法

通过引入变换和选准试探函数,把难于求解的非线性偏微分方程化为易于求解的代数方程,然后用待定系数法确定相应的常数,从而简洁地求得了非线性薛定谔方程的解析解.     

An asymptotic solving method for the periodic solution of a class of disturbed nonlinear evolution equation

<正>A class of disturbed evolution equation is considered using a simple and valid technique.We first introduce the periodic traveling-wave solution of a corresponding typical evolution equation.Then the approximate solution for an original disturbed evolution equation is obtained using the asymptotic method.We point out that the series of approximate solution is convergent and the accuracy of the asymptotic solution is studied using the fixed point theorem for the functional analysis.     

Two alternative methods for solving ion transport equation with anisotropic approximation

By means of two alternative methods namely, the maximum entropy and Chapman-Enskog, flux limited approach the ion transport equation for slowing-down problem of low-energy light ions in solid has been solved explicitly. Maximum entropy technique yields approximate solutions in the form of locally Maxwellian distribution function, based on moments expansion truncated upon entropy maximization.The behavior of the approximate maximum entropy and the flux limited solutions have the same tendency. Knowing the distribution function obtained by flux limited approach, allows us to calculate directly the path length distributions of backscattered ions, and compared with that found by other theories such as Laplace-transform and double Legendre polynomial approximation. One can see that the flux limited approach is better than the previous method namely, (DPN) Laplace-transform.The results reported in this article provide further evidence of the usefulness of both maximum entropy and flux limited for obtaining the solution of ion transport equation in compact form.     

Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation

In this paper, we studied time-fractional nonlinear partial differential equations to reach their some solutions. There are lots of explicit and analytic methods in the literature. We used Kudryashov, Exp-function, and Jacobi elliptic rational expansion methods. By using these methods, we get some solutions of time-fractional fifth-order KdV-like equation.     

The constructive technique and its application in solving a nonlinear reaction diffusion equation

A mathematical technique based on the consideration of a nonlinear partial differential equation together with an additional condition in the form of an ordinary differential equation is employed to study a nonlinear reaction diffusion equation which describes a real process in physics and in chemistry. Several exact solutions for the equation are acquired under certain circumstances.     

求解非线性传输线方程的简便方法

应用sine-cosine方法构造了非线性传输线方程的周期波解和孤波解.结果表明,sine-cosine方法是一种简便、有效的方法,并且可以用于求解其他非线性方程.     

Klein-Gordon方程初边值问题的一种新的差分方法

对非线性Kiein-Gordon方程的初边值问题提出了一种能量守恒差分格式。证明了该格式的收敛性和稳定性。并给出数值计算结果。     

Numerical solution of the conformable space-time fractional wave equation

In this paper, an efficient numerical method is considered for solving space-time fractional wave equation. The fractional derivatives are described in the conformable sense. The method is based on shifted Chebyshev polynomials of the second kind. Unknown function is written as Chebyshev series with the N term. The space-time fractional wave equation is reduced to a system of ordinary differential equations by using the properties of Chebyshev polynomials. The finite difference method is applied to solve this system of equations. Numerical results are provided to verify the accuracy and efficiency of the proposed approach.