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In this article, the (3 + 1)-dimensional generalized Bogoyavlensky–Konopelchenko equation has been examined for finding its exact closed form solitonic solutions with the help of symmetries. These symmetries are investigated by the means of the Lie symmetry approach, which is also known as classical Lie group method and geometric approach. In the geometric approach, the extended Harrison and Estabrook's differential forms have been used to investigate the infinitesimals of the generalized Bogoyavlensky–Konopelchenko equation. Moreover, by using formal Lagrangian, it is proved that the aforementioned partial differential equation satisfies the quasi-self-adjointness condition. Additionally, the conservation laws for the (3 + 1)-dimensional generalized Bogoyavlensky–Konopelchenko equation have been derived by imposing the "new conservation theorem," which was proposed by Ibragimov. Finally, the exact closed form solutions are obtained with the help of Lie symmetries corresponding to the defined model. [ABSTRACT FROM AUTHOR]

In this paper, the local discontinuous Galerkin method is used to analyze numerical solutions for nonlinear Allen–Cahn equations with nonperiodic boundary conditions. To begin with, the spatial variables are discretized to generate a semidiscrete method of lines scheme. This yields an ordinary differential equation system in the temporal variable, which is then solved using the higher-order total variation diminishing Runge–Kutta method. A comparison of the generated numerical results to the exact results for various test problems using different tables and figures provides insight into the effectiveness and accuracy of the proposed method. The numerical results confirm that the proposed method is an effective numerical scheme for solving the Allen–Cahn equation since the obtained solutions are extremely close to the exact solutions while exhibiting substantially less error. [ABSTRACT FROM AUTHOR]

The governing equations for fluid flows, i.e. Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) model equations represent a water wave model. These model equations describe the bidirectional propagating water wave surface. In this paper, an auto-Bäcklund transformation is being generated by utilizing truncated Painlevé expansion method for the considered equation. This paper determines the new bright soliton solutions for (2 + 1) and (3 + 1) -dimensional nonlinear KP-BBM equations. The simplified version of Hirota's technique is utilized to infer new bright soliton solutions. The results are plotted graphically to understand the physical behavior of solutions. [ABSTRACT FROM AUTHOR]

In this article, the space-time fractional perturbed nonlinear Schrödinger equation (NLSE) in nanofibers is studied using the improved tan (ϕ (ξ) / 2) expansion method (ITEM) to explore new exact solutions. The perturbed nonlinear Schrodinger equation is a nonlinear model that occurs in nanofibers. The ITEM is an efficient method to obtain the exact solutions for nonlinear differential equations. With the help of the modified Riemann–Liouville derivative, an equivalent ordinary differential equation has been obtained from the nonlinear fractional differential equation. Several new exact solutions to the fractional perturbed NLSE have been devised using the ITEM, which is the latest proficient method for analyzing nonlinear partial differential models. The proposed method may be applied for searching exact travelling wave solutions of other nonlinear fractional partial differential equations that appear in engineering and physics fields. Furthermore, the obtained soliton solutions are depicted in some 3D graphs to observe the behaviour of these solutions. [ABSTRACT FROM AUTHOR]

In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, the symmetry analysis and conservation laws of time fractional coupled Ablowitz–Kaup–Newell–Segur (AKNS) equations have been presented. Initially, the Lie symmetry method has been used for similarity reduction of time fractional coupled AKNS equations. Here, the Erdélyi–Kober differential operators have been used for symmetry reduction. Also, the new conservation vectors for time fractional coupled AKNS equations have been derived by using the new conservation theorem with formal Lagrangian. [ABSTRACT FROM AUTHOR]

In this paper, we have studied space-time Brownian motion and its applications to mixed type stochastic integral equations. Approximate solutions of mixed stochastic integral equations have been obtained by using two-dimensional (2D) second kind Chebyshev wavelets (CWs). Furthermore, some examples have been presented to justify the efficiency of 2D second kind CWs. [ABSTRACT FROM AUTHOR]

Purpose: A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials. Design/methodology/approach: To solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations. Findings: Numerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate. Originality/value: The efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations. [ABSTRACT FROM AUTHOR]