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This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion. The sine-Gordon expansion and the hyperbolic function approaches are efficiently applied to the family of Boussinesq-like equations to explore novel solitary, kink, anti-kink, combo, and singular-periodic wave solutions. The attained solutions are expressed by the trigonometric and hyperbolic functions including tan , sec , cot , csc , tanh , sech , coth , csch , and of their combination. In addition, the mentioned two approaches are applied to the aforesaid models in the sense of Atangana conformable derivative or Beta derivative to attain new wave solutions. Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying the corresponding equations of interest are provided to understand the underlying mechanisms of the stated family. For the bright wave solutions in terms of Atangana’s conformable derivative, the amplitudes of the bright wave gradually decrease, but the smoothness increases when the fractional parameters α and β increase. On the other hand, the periodicity increases for the periodic wave solutions. The attained new wave solutions can motivate applied scientists for engineering their ideas to an optimal level and they can be used for the validation of numerical simulation results in the propagation of waves in shallow water and other nonlinear cases. The performed approaches are found to be simple and efficient enough to estimate the solutions attained in the study and can be used to solve various classes of nonlinear partial differential equations arising in mathematical physics and engineering.

In order to compare different solvers for systems of nonlinear equations, some novel goodness and qualification criteria are defined in this paper. These use all parameters of a nonlinear solver such as convergence order, number of function evaluations, number of iterations, CPU time, etc. To achieve the criteria, different algorithms to solve nonlinear systems are categorised to three kinds. For any category, two criteria are defined to compare different algorithms in that category. As numerical results show, these new criteria can use to compare different algorithms which solve systems of nonlinear equations. Further, we present some corrected formulas for some classical efficiency indices and change them to be more applicable. Also, some suggestions are presented about the future works.

In this paper, in order to solve systems of nonlinear equations, a new class of frozen Jacobian multi-step iterative methods is presented. Our proposed algorithms are characterized by a highly convergent order and an excellent efficiency index. The theoretical analysis is presented in detail. Finally, numerical experiments are presented for showing the performance of the proposed methods, when compared with known algorithms taken from the literature.

This article is devoted to a new semi-analytical algorithm for solving time-fractional modified anomalous sub-diffusion equations (FMASDEs). In this method first, the main problem is reduced to a system of fractional-order ordinary differential equations (FODEs) under known initial value conditions by using the Chebyshev collocation procedure. After that, to solve this system, some auxiliary initial value problems are defined. Next, we find an optimal linear combination of some particular solutions for these problems and finally we use this linear combination to construct a semi-analytical approximate solution for the main problem. To demonstrate the convergence property of the new method, a residual error analysis is performed in details. Some test problems are investigated to show reliability and accuracy of the proposed method. Besides, convergence order's indicators are evaluated for all test problems and are compared with ones of the other methods. Moreover, a comparison between our computed numerical results and the reported results of the other numerical schemes in the literature exhibits that the proposed technique is more precise and reliable. In summary advantages of the proposed method are: high accuracy, easy programming, high experimental convergence order, and solving another types of fractional differential equations.

This paper presents, a novel semi-analytical algorithm, based on the Chebyshev collocation method, for the solution of space-fractional diffusion equations. The original fractional equation is transformed into a system of ordinary differential equations (ODEs) by the Chebyshev collocation method. A new semi-analytical method is then used to approximate the solution of the resulting system. To emphasize the reliability of the new scheme, a convergence analysis is presented. It is shown that highly accurate solutions can be achieved with relatively few approximating terms and absolute errors are rapidly decrease as the number of approximating terms is increased. By presenting some numerical examples, we show that the proposed method is a powerful and reliable algorithm for solving space-fractional diffusion equations and can be extended to solve another space-fractional partial differential equations. Comparison with other methods in the literature, demonstrates that the proposed method is both efficient and accurate.

A new semi-analytical algorithm is presented to solve general multi-point boundary value problems. This method can be applied on nth order linear, nonlinear, singular and nonsingular multi-point boundary value problems. Mathematical base of the method is presented; convergence of the method is proved. Also, the algorithm is applied to solve multi-point boundary value problems including nonlinear sixth-order, nonlinear singular second-order five-point boundary value problems, and a singularly perturbed boundary value problem. Comparison results show that the new method works more accurate than the other methods.

This paper is devoted to the application of the method of lines to solve one-dimensional diffusion equation where the classical (integer) second derivative is replaced by a fractional derivative of the Caputo type of order α less than 2 as the space derivative. A system of initial value problems approximates the solution of the fractional diffusion equation with spline approximation of the Caputo derivative. The result is a numerical approach of order O ( Δ x 2 + Δ t m ) , where Δx and Δt denote spatial and temporal step-sizes, and 1 ≤ m ≤ 5 is an integer which is set by an ODE integrator that we used. The convergence and numerical stability of the method are considered, and numerical tests to investigate the efficiency and feasibility of the scheme are provided.

In this paper, we address the propagation of optical solitons through a magneto-electro-elastic media. We apply a variety of methods, namely the semi-inverse variational principle, sine–cosine func...

[EN] In this paper, a new technique to construct a family of divided differences for designing derivative-free iterative methods for solving nonlinear systems is proposed. By using these divided differences any kind of iterative methods containing a Jacobian matrix in its iterative expression can be transformed into a "Jacobian-free" scheme preserving the order of convergence. This procedure is applied on different schemes, showing theoretically their order and error equation. Numerical experiments confirm the theoretical results and show the efficiency and performance of the new Jacobian-free schemes. (C) 2018 Published by Elsevier B.V., This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P, MTM2015-64013-P and Generalitat Valenciana PROMETEO/2016/089.

In this paper, we propose two new extensions of the rational trigonometric methods for solving distinct forms of Boussinesq-like equations. The proposed methods give solutions which are expressed b...

Space-time conformable fractional nonlinear ( 1 + 1 )-dimensional Schrodinger-type models are investigated in this paper. Traveling wave solutions using the sine-Gordon expansion approach for these models are presented. The sine-Gordon expansion method is used to obtain exact solutions for three types of space-time conformable fractional nonlinear Schrodinger-type equations which some of them are new.

We investigate the corrected entropy and Hawking temperature of the BTZ black hole which obtained from (2 + 1) dimensional black hole. Besides, we generalize our analysis of black holes to the case of Friedmann-Robertson-Walker (FRW) universe. The corrections to the Hawking temperature and entropy of apparent horizon for FRW universe are also obtained. Comparing the results with the high energy black hole demonstrates how the semi-classic approximation affects the thermodynamics of the BTZ black hole, corrected terms, classical action and the entropy.

In this paper, we obtain traveling wave solutions for time-fractional equations. The homotopy perturbation method is used to obtain these solutions. It is shown that, these solutions have continuity dependence on the time fractional derivatives.

In this work, we present a semi-analytical method for solving integro-differential equations under multi-point or mixed boundary conditions. The proposed method solves linear and nonlinear FredholmVolterra integro-differential equations. A convergence analysis of the proposed method is directly examined. Numerical examples are worked out to demonstrate the main results. Moreover, proper graphs are provided to confirm the efficiency and the accuracy of the proposed scheme. We show that with a few number of obtained approximating terms, we achieve a high accuracy level of the obtained results. However, increasing the number of approximating terms, yields a significant decrease of the error of the approximation. The proposed method is very useful, reliable, and flexible for solving different kinds of integro-differential equations.

In this work we present a semi-analytical method to solve systems of integro-differential equations under mixed boundary conditions. The proposed method handles linear and nonlinear systems of FredholmVolterra integro-differential equations in a reliable manner. We present a convergence analysis for the proposed method to emphasize its reliability. Numerical examples are examined to show the efficiency and the accuracy of the scheme. We show that a few number of approximating terms gives results of high accuracy level, and by increasing the number of these terms, the error of the approximated solution decreases rapidly. The proposed method is powerful and reliable to solve other kinds of systems of integro-differential equations. Further, it has a small CPU time and its computational time is less than the other methods. A new method to solve systems of IDEs under mixed boundary conditions is presented.This algorithm can solve different kinds of systems of integro-differential equations.Convergence analysis for the method is proved and accuracy of the method is high.Accuracy of results is truly fine even if the number of approximating terms is small.CPU time and computational time of the new method is less than the other methods.

In this paper, an efficient numerical method for the solution of fractional nonlinear (1 + 1)-dimensional Biswas–Milovic equation based on the homotopy perturbation method is presented. Two kinds of fractional differentiation, namely space- and space–time-fractional ones are considered. The Caputo sense of fractional derivatives are investigated. Application of homotopy perturbation method shows that this method is a promising algorithm to solve wide classes of nonlinear fractional order partial differential equations with complex solutions.

In this paper, the so-called Adomian decomposition method is used to obtain semi-analytical solutions for various types of nonlinear fractional Biswas–Milovic equation. These semi-analytical solutions are obtained for space–time-fractional versions of the equation. As figure plots show, these solutions are multi-wave ones. There is a reasonable agreement between these solutions.

This paper addresses the soliton solutions for a variety of Boussinesq-like equations with spatio-temporal dispersion. The semi-inverse variational principle (SVP) is applied to obtain these solutions. The derived solutions address the dynamics of thin inviscid layers with free surface, solitons solutions, and other nonlinear phenomena. The structures of the obtained solutions were carried out and illustrated, and can provide feasible basis for studying surface water models.

In this paper, we modify the idea of Haar wavelets method to obtain semi-analytical solutions for the systems of three-dimensional nonlinear partial differential equations. Theoretical considerations are discussed. To demonstrate the efficiency of the method, a test problem is presented. The approximate solutions of the system of three-dimensional nonlinear partial differential equations are compared with the exact solutions as well as existing numerical solutions found in the literature. The numerical solutions which are obtained using the suggested method show that numerical solutions are in a very good coincidence with the exact solutions.

An accurate semi-analytical method to solve fractional Riccati differential equation (FRDE) with constant coefficients is presented. We predict some properties of the fractional derivative of solution of an FRDE and by introducing a semi-analytical method its solution is obtained. This solution has all properties which we have predicted. Accuracy and efficiency of the presented method are shown by solving some examples. Comparison with the other methods shows that our method works very well. Further, numerical stability investigation shows that the presented semi-analytical method is a stable algorithm.

[EN] It is well known that scalar iterative methods with derivatives are highly more stable than their derivative-free partners, understanding the term stability as a measure of the wideness of the set of converging initial estimations. In multivariate case, multidimensional dynamical analysis allows us to afford this task and it is made on different Jacobian-free variants of Newton¿s method, whose estimations of the Jacobian matrix have increasing order. The respective basins of attraction and the number of fixed and critical points give us valuable information in this sense., This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.

[EN] A new HSS-based algorithm for solving systems of nonlinear equations is presented and its semilocal convergence is proved. Spectral properties of the new method are investigated. Performance profile for the new scheme is computed and compared with HSS algorithm. Besides, by a numerical example in which a two-dimensional nonlinear convection diffusion equation is solved, we compare the new method and the Newton-HSS method. Numerical results show that the new scheme solves the problem faster than the NewtonHSS scheme in terms of CPU -time and number of iterations. Moreover, the application of the new method is found to be fast, reliable, flexible, accurate, and has small CPU time., This research was partially supported by Ministerio de Economia y Competitividad, Spain under grants MTM2014-52016-C2-2-P and Generalitat Valenciana, Spain PROMETEO/2016/089.

In this article, we introduce a family of nonlinear (1+1) dimensions Schrödinger-type models with space-time fractional evolution in the sense of a conformable fractional derivative. We apply the modified Kudryashov method in context of fractional complex transformation and seek a variety of optical soliton solutions for these equations. The modified Kudryashov method is efficient and consistent for solving nonlinear space-time fractional differential equations.

Purpose The purpose of this paper is to take the thermal analysis of natural convection and radiation heat transfer in fully wet porous fins. The wet porous fins taken for the analysis are straight fins in nature and wet. Their profile being straight helps heat transfer process of fins faster. The analysis is performed using the Darcy’s model to generate the heat equation to analyze the variation of convection and radiation parameters. The porous nature of the fins allows the flow to penetrate through the porous material of the fins leading to solid-fluid interface. The obtained non-dimensional ordinary differential equation involving three highly nonlinear terms are solved numerically by using spectral collocation method after which they are reduced into algebraic equations using Chebyshev polynomials. The study is analyzed using the mathematical analysis on heat equation and generating graphs for finding the parameters important to the heat transfer in the straight fins. Design/methodology/approach This study is performed using Darcy’s model to formulate heat transfer equation. To study the thermal performance, the authors considered a finite length fin with insulated tip. The effects of the wet fin parameter m2, porosity parameter Sh, radiation parameter G and temperature ratio CT on the dimensionless temperature distribution and heat transfer rate are discussed. Findings The results show that the base heat flow increases when the permeability of the medium is high and/or when the buoyancy effect induced in the fluid is strong. Research limitations/implications The analysis is made for the Darcy’s model. Non-Darcy effects will be investigated in a future work. Practical implications The approach is useful in enhancing heat transfer rates. Originality/value The results of the study will be interest to the researchers of the field of heat exchanger designers.

This work deals with fractional version of (1 + 1)-dimensional Biswas–Milovic equation that describes the long-distance optical communications. Some soliton and traveling wave solutions are obtained for the equation by means of four trigonometric analytical methods and symbolic computations. The solution of the problem by means of these methods reduces the independent variables in the fractional equation by one leading to nonlinear ordinary differential equations. The resulting nonlinear ordinary differential equations are then solved analytically using Maple software. All of obtained solutions are new for the fractional Biswas–Milovic equation. The work shows the power of these schemes and the variety of the obtained solutions.

This paper is concerned with extension of some analytical methods to obtain exact solutions of nonlinear fractional (1 + 1)-dimensional symmetric regularized long wave (SRLW) equation. These methods are straightforward, concise, effective and easy to use. They are powerful mathematical tools to obtain exact solutions of nonlinear fractional differential equations and can be used to solve another nonlinear fractional equations.

The present study introduces a general form of fractional Biswas–Milovic (FBM) equation. After this, some exact solutions for FBM equation are obtained by Exp-function and sine–cosine methods. This work illustrates the validity and great potential of the Exp-function and sine–cosine approaches for the nonlinear space–time fractional differential equations with complex valued solutions. One can see that the methods are relatively easy and efficient to use.

In this paper, the first integral method and a variety of solitary waves methods are employed for constructing the new exact traveling wave solutions of a (2 + 1)-dimensional Zakharov-like soliton equation. Moreover, this equation will be studied by a variety of solitary waves methods and ansatze.

The multiple exp-function method is a new approach to obtain multiple-wave solutions of nonlinear partial differential equations (NLPDEs). By this method, one can obtain multi-soliton solutions of NLPDEs. Hence, in this paper, using symbolic computation, we apply the multiple exp-function method to construct the exact multiple-wave solutions of a (3 + 1)-dimensional soliton equation. Based on this application, we obtain mobile single-wave, double-wave and multi-wave solutions for this equation. In addition, we employ the straightforward and algebraic Hirota bilinearization method to construct the multi-soliton solutions of NLPDEs, and we reveal the remarkable property of soliton–soliton collision through this approach. Further, we investigate the one- and two-soliton solutions of a (3 + 1)-dimensional soliton equation using the Hirota’s method. We explore the particle-like behavior or elastic interaction of solitons, which has potential application in optical communication systems and switching devices.

In this paper, we discretize time derivative terms by a forward difference scheme and linearize the nonlinear terms using a quasilinearization technique to reduce the original equation into a system of ordinary differential equations. Then the Haar wavelet quasilinearization approach is applied to compute the numerical solutions of the Hunter–Saxton equation. Computer simulations show that our obtained results are in a good agreement with the exact solution.

In this paper, Haar wavelets method (HWM) is applied to compute the numerical solutions of the foam drainage equation. The mathematical base of the method is presented. First, the time derivative is discretized by a forward difference scheme and then a quasilinearization technique is used to linearize the foam drainage equation. Finally, we solve a system of linear equations which is obtained by applying the Haar wavelet method for discretizing the space derivatives. Obtained results by HWM are very similar to the exact solutions. Further, a comparison between our results and results which are obtained by HPM, HPTM, LDM and ADM is presented. Numerical results show that our method works better than the other methods.

Heat transfer with natural convection and radiation effect on a fully wet porous radial fin is considered. The radial velocity of the buoyancy driven flow at any radial location is obtained by applying Darcy’s law. The obtained non-dimensionalized ordinary differential equation involving three highly nonlinear terms is solved numerically with the spectral collocation method. In this approach, the dimensionless temperature is approximated by Chebyshev polynomials and discretized by Chebyshev-Gausse-Lobatto collocation points. A particular algorithm is used to reduce the nonlinearity of the conservation of energy equation. The present analysis characterizes the effect of ambient temperature in different ways and it provides a better picture regarding the effect of ambient temperature on the thermal performance of the fin. The profiles for temperature distributions and dimensionless base heat flow are obtained for different parameters which influence the heat transfer rate.

An annular fin of hyperbolic profile with temperature dependent thermal conductivity is studied by pseudospectral method. Graphs illustrating the effect of fin dimensions, surface convection characteristics and the thermal conductivity parameter on the thermal performance of the fin are presented and discussed. A comparison of the obtained numerical results is made with the closed form analytical solution available in the literature for the case of constant thermal conductivity. This comparison confirms the high accuracy of numerical results. When the thermal conductivity increases with temperature, the effect is to elevate both the temperature distribution in the fin and the fin efficiency. The converse is true when the thermal conductivity decreases with temperature.

In this paper, we study three different space-time fractional models of the Schrodinger equation. By using the properties of conformable derivative and fractional complex transform, the bright, dark and singular optical solitons for conformable space–time fractional nonlinear $$(1+1)$$ -dimensional Schrodinger models are determined.

The present study emphasis to look for new closed form exact solitary wave solutions for the variety of fractional Boussinesq-like equations using the modified Kudryashov method with the help of symbolic computation. As a consequence, the modified Kudryashov method is successfully employed and acquired some new exact solitary wave solutions in terms of exponential based functions with fractional version. All solutions have been verified back into its corresponding equation with the aid of Maple package program. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed method is robust and efficient than other methods and the obtained solutions in this paper can help us to understand the variation of solitary waves in the field of oceanography.

In this paper, the sine–cosine method is employed to construct exact solutions of the space-time fractional ( $$1+1$$ )-dimensional nonlinear Schrodinger models. Many new families of exact traveling wave solutions of these models are successfully obtained. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear space-time fractional evolution equations in mathematical physics.

We explore the shape changing and clevaging nature of anti-kink solutions of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation. We achieved this by invoking the multiple exp-function method aided with symbolic computation which remains an indispensable tool to deal with computational algebraic systems.

An analytic (series) solution is developed to describe the thermal performance of a porous radial fin with natural convection in the fluid saturating the fin and radiation heat loss from the top and bottom surfaces of the fin. The HAM results for the temperature distribution and base heat flux are compared with the direct numerical results and found to be very accurate.

In this paper, we study the flow of a micropolar fluid in a porous channel with expanding or contracting walls. First, we use spectral collocation method on the governing equations to obtain an initial approximation for the solution of equations. Then using the obtained initial approximation, we apply the homotopy analysis method to obtain a recursive formula for the solution.

In this study, the effects of transient thermal performance of a rectangular porous fin in the presence of radiation and natural convection heat transfer are considered. The porous fin allows the flow to infiltrate through it and solid–fluid interaction takes place. This study is performed using Darcy’s model to formulate heat transfer equation. To study the thermal performance, three types of cases are considered, namely, long fin, finite length fin with insulated tip and finite length fin with tip exposed. The effects of the porosity parameter Sh, radiation parameter G and the temperature ratio CT on the dimensionless transient temperature distribution and heat transfer rate are discussed.

In this study, the effects of radiation and convection heat transfer in a radial porous fin are considered. The geometry considered is that of a rectangular profile fin. The porous fin allows the flow to infiltrate through it and solid-fluid interaction takes place. This study is performed using Darcy’s model to formulate the heat transfer equation. The thermal conductivity is assumed to be a function of temperature. The effects of the natural convection parameter Nc , radiation parameter Nr and thermal conductivity parameter m on the dimensionless temperature distribution and heat transfer rate are discussed. The results suggest that the radiation transfers more heat than a similar model without radiation

Purpose – The purpose of this paper is to conduct a numerical study of the convection heat transfer in porous media by the homotopy analysis method (HAM). The geometry considered is that of a rectangular profile fin. The porous fin allows the flow to infiltrate through it and solid-fluid interaction takes place. This study is performed using Darcy's model to formulate heat transfer equation. To study the thermal performance, three types of cases are considered namely long fin, finite length fin with insulated tip and finite length fin with tip exposed. The theory section addresses the derived governing equation. The effects of the porosity parameter Sh, radiation parameter G and temperature ratio CT on the dimensionless temperature distribution and heat transfer rate are discussed. The results suggest that the radiation transfers more heat than a similar model without radiation. The auxiliary parameter in the HAM is derived by using the averaged residual error concept which significantly reduces the computational time. The use of optimal auxiliary parameter provides a superior control on the convergence and accuracy of the analytic solution. Design/methodology/approach – This study is performed using Darcy's model to formulate heat transfer equation. To study the thermal performance, three types of cases are considered namely long fin, finite length fin with insulated tip and finite length fin with tip exposed. The effects of the porosity parameter Sh, radiation parameter G and temperature ratio CT on the dimensionless temperature distribution and heat transfer rate are discussed. Findings – The HAM has been successfully applied for the thermal performance of a porous fin of rectangular profile. Solutions are derived for three cases of tip condition: an infinitely long fin with tip in thermal equilibrium with the ambient, a finite fin with an insulated tip and a finite fin with a convective tip. The performance of the fin depends on three dimensionless parameters; porosity parameter Sh, radiation-conduction parameter G and a dimensionless temperature relating the ambient and base temperatures. The results show that the base heat flow increases when the permeability of the medium is high and/or when the buoyancy effect induced in the fluid is strong. The base heat flow is enhanced as the surface radiation or the tip Biot number increases. Research limitations/implications – The analysis is made for the Darcy's model. Non-Darcy effects will be investigated in a future work. Practical implications – The approach is useful in enhancing heat transfer rates. Originality/value – The results of the study will be interested to the researchers of the field of heat exchanger designers.

We study the effects of the generalized uncertainty principle in the tunneling formalism for Hawking radiation to evaluate the quantum-corrected Hawking temperature and entropy for a Kerr black hole. By assumption of a spatially flat universe accompanied with expansion of metric, the modified area and entropy of Kerr black hole are calculated and we could obtain an expression for entropy of black hole that is changing with respect to time and Bekenstein-Hawking temperature.

The properties of quintom model are investigated in the isotropic and homogeneous universe as a dynamical system dominated by dark energy including the phantom and quintessence fields. A general discussion about the phase space of spatially non-flat universe is presented. We study the results for the later times without assuming the specific form of the potential. Then, we exhibit an obvious structure for the dynamics of the system.

It is shown that the Bekenstein-Hawking entropy of black holes can accept a correction that effects on the radiation tunneling probability. By assumption of a spatially flat universe accompanied with expansion of metric, we could obtain an expression for entropy of black hole that is changing with respect to time and Bekenstein-Hawking temperature.