New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

The pivotal aim of the present work is to find the numerical solution for fractional Benney–Lin equation by using two efficient methods, called q ‐homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three‐dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology.     

On Somos’s theta-function identities of level 14

M. Somos discovered around 6200 theta-function identities of different levels using the computer and offered no proof for them. The purpose of this paper is to prove twenty four of his identities of level 14.     

New approach for fractional Schrödinger-Boussinesq equations with Mittag-Leffler kernel

In this paper, we find the solution and analyse the behaviour of the obtained results for the nonlinear Schrödinger-Boussinesq equations using q -homotopy analysis transform method (q -HATM) within the frame of fractional order. The considered system describes the interfaces between intermediate long and short waves. The projected fractional operator is proposed with the help of Mittag-Leffler function to incorporate the nonsingular kernel to the system. The projected algorithm is a modified and accurate method with the help of Laplace transform. The convergence analysis is presented with the help of the fixed point theorem in the form existence and uniqueness. To validate and illustrate the effectiveness of the algorithm considered, we exemplified considered system with respect of arbitrary order. Further, the behaviour of achieved results is captured in contour and 3D plots for distinct arbitrary order. The results show that the projected scheme is very effective, highly methodical and easy to apply for complex and nonlinear systems and help us to captured associated behaviour diverse classes of the phenomenon.     

An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model

In this paper, the q -homotopy analysis transform method (q -HATM) is applied to find the solution for the fractional Lakshmanan-Porsezian-Daniel (LPD) model. The LPD model is the generalization of the non-linear Schrödinger (NLS) equation. The proposed method is graceful fusions of Laplace transform technique with q -homotopy analysis scheme, and the derivative is considered in Caputo sense. In order to validate and illustrate the efficiency of the proposed method, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured for the three different cases in terms of 3D and contour plots for diverse values of the fractional order. The obtained results confirm that the future method is easy to implement, highly methodical, and very effective to analyse the behaviour of complex non-linear fractional differential equations exist in the connected areas of science and engineering.     

Novel approach for modified forms of Camassa-Holm and Degasperis-Procesi equations using fractional operator

In the present study, we consider the q-homotopy analysis transform method to find the solution for modified Camassa-Holm and Degasperis-Procesi equations using the Caputo fractional operator. Both the considered equations are nonlinear and exemplify shallow water behaviour. We present the solution procedure for the fractional operator and the projected solution procedure gives a rapidly convergent series solution. The solution behaviour is demonstrated as compared with the exact solution and the response is plotted in 2D plots for a diverse fractional-order achieved by the Caputo derivative to show the importance of incorporating the generalised concept. The accuracy of the considered method is illustrated with available results in the numerical simulation. The convergence providence of the achieved solution is established in $\hslash $-curves for a distinct arbitrary order. Moreover, some simulations and the important nature of the considered model, with the help of obtained results, shows the efficiency of the considered fractional operator and algorithm, while examining the nonlinear equations describing real-world problems.     

Analysis of fractional Swift-Hohenberg equation using a novel computational technique

In this paper, the approximated analytical solution for fractional Swift-Hohenberg (S-H) equation is found with the aid of novel technique called q -homotopy analysis transform method (q -HATM) . To ensure the applicability and efficiency of the proposed algorithm, we consider non-linear arbitrary-order S-H equation in presence and absence of dispersive term. The convergence analysis for the projected problem is presented, and the numerical simulations have been conducted to verify the future scheme is reliable and accurate. Further, the effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are presented through plots. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyse the complex problems that arose in science and technology.     

Partition identities arising from Somos’s theta function identities

In this paper, we establish certain partition identities for theta function identities of level 14 discovered by M. Somos.     

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

The aim of the present work is to find the numerical solutions for time‐fractional coupled Burgers equations using a new novel technique, called fractional natural decomposition method (FNDM). Two examples are considered in order to illustrate and validate the efficiency of the proposed algorithm. The numerical simulation has been conducted to ensure the exactness of the present method, and the obtained solutions are offered graphically to reveal the applicability and reliability of the FNDM. The outcomes of the study reveal that the FNDM is computationally very effective and accurate to study the (2 + 1)‐dimensional coupled Burger equations of arbitrary order.     

Novel Synthesis of 5‐Substituted‐3‐phenylindole‐2‐(1,2,4‐triazole) Derivatives

Various substituted 3‐phenylindole 2‐carboxylates (1a–c) were prepared according to the literature methods. These carboxylates (1a–c) on reaction with thiosemicarbazide yielded 5‐substituted‐3‐phenylindol‐2‐(1,2,4‐triazole‐3‐thione) (2a–c) on refluxing in pyridine for 8 h. The 5‐substituted‐3‐phenylindole‐2‐[1,2,4‐triazolo‐3‐thioacetic acid] (3a–c) were prepared from 5‐substituted‐3‐phenyl indole‐2‐[1,2,4‐triazole‐3‐thione] (2a–c) on reaction with an appropriate alkylating agent and sodium acetate in acetic acid. Further, (3a–c) were reacted with acetic anhydride to bring about a cyclocondensation reaction to yield 5‐substituted‐3‐phenylindol‐2‐thiazolo(2,3‐b)‐triazole (4a–c). The 5‐substituted‐3‐phenylindole‐2‐[1,2,4‐triazolo‐3‐acetic acid] (3a–c) were reacted with o‐phenylenediamino dihydrochloride in ethylene glycol to yield 5‐substituted‐3‐phenylindole‐1,2,4‐triazolo‐3′‐yl‐thiomethyl)benzimidazoles (5a–c).