Explore the vast range of titles available.

In this paper, we use the modified exp−ψθ-function method to observe some of the solitary wave solutions for the microtubules (MTs). By treating the issues as nonlinear model partial differential equations describing microtubules, we were able to solve the problem. We then found specific solutions to the nonlinear evolution equation (NLEE) covering various parameters that are particularly significant in biophysics and nanobiosciences. In addition to the soliton-like pulse solutions, we also find the rational, trigonometric, hyperbolic, and exponential function characteristic solutions for this equation. The validity of the method we developed and the fact that it provides more solutions are demonstrated by comparison to other methods. We next use the software Mathematica 10 to generate 2D, 3D, and contour plots of the precise findings we observed using the suggested technique and the proper parameter values.

We investigate a (3 + 1)-dimensional nonlinear evolution equation which is a higher-dimensional generalization of the Korteweg–de Vries equation. On the basis of the decomposition approach, the N -antidark soliton solution on a finite background is constructed by using the Darboux transformation together with the limit technique. The asymptotic analysis for the N -antidark soliton solution is performed, and the collision between multiple antidark solitons is proved to be elastic. Under the velocity resonant mechanism, the antidark soliton molecules on the ( x ,  t ), ( y ,  t ), ( y ,  z ) and ( t ,  z ) planes are found instead of the ( x ,  y ) and ( x ,  z ) planes. Based on the three- and the four-antidark soliton solutions, the elastic collision between a soliton molecule and a common soliton and the elastic collision between two soliton molecules are analytically demonstrated, respectively. These results may be useful for the study of soliton molecules in hydrodynamics and nonlinear optics.

This paper aims at computing M-lump solutions for the ( 3 + 1 ) -dimensional nonlinear evolution equation. These solutions in all directions decline to an identical state obtained by employing the “long wave” limit with respect to the N-soliton solutions which are got by using the direct methods. Subsequently, we discuss the dynamic properties of the M-lump solutions which describe the multiple collisions of lumps. Based on the obtained lump solutions, the lump–kink solutions are also obtained. In addition, the periodic interactive solutions are given.

The present paper presents an abstract theory for proving (local-in-time) existence of energy solutions to some doubly-nonlinear evolution equations of gradient flow type involving time-dependent subdifferential operators with a quantitative estimate for the local-existence time. Furthermore, the abstract theory is employed to obtain an optimal existence result for some doubly-nonlinear parabolic equations involving space-time variable exponents, which are (possibly) non-monotone in time. More precisely, global-in-time existence of solutions is proved for the parabolic equations.

In this article we propose efficient techniques for solving fractional differential equations such as KdV-Burgers, Kadomtsev-Petviashvili, ZakharovKuznetsov with less computational efforts and high accuracy for both numerical and analytical purposes. The general expa-function method is employed to reckon new exact solitary wave solutions of time fractional nonlinear evolution equations (NLEEs) stemming from mathematical physics. Fractional complex transformation in conjunction with modified Riemann-Liouville operator is used to tackle the fractional sense of the accompanying problems. A comparison with existing conventional exp-function method and improved exp-function method shows that the proposed recipe is more productive in terms of obtaining analytical solutions. The graphical depictions of extracted information show a strong relationship among fractional order outcomes with those of classical ones.

Long description: This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by nonlinear elliptic partial differential equations (PDEs). To iteratively solve such BVPs, it is of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. The new adaptive wavelet theory guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the ell₂ sequence spaces of expansion coefficients exist. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs. Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of nonlinear PDE sub-problems. This thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve nonlinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory.

The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces is established. The strong convergence of this technique is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to variational inequalities, hierarchical minimization problems, and nonlinear evolution equations are included.

A number of solitary wave solutions for microtubules (MTs) are observed in this article by using the modified exp-function approach. We tackle the problem by treating the results as nonlinear RLC transmission lines, and then finding exact solutions to Nonlinear Evolution Equation (NLEE) containing parameters of particular importance in biophysics and nanobiosciences. For this equation, we find trigonometric, hyperbolic, rational, and exponential function solutions, as well as soliton-like pulse solutions. A comparison with other approach indicates the legitimacy of the approach we devised as well as the fact that our method offers extra solutions. Finally, we plot 2D, 3D and contour visualizations of the exact results that we observed using our approach using appropriate parameter values with the help of software Mathematica 10.

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models.