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In science and technology, the phenomena of transportation are crucial. Advection and diffusion can occur in a wide range of applications. Distinct types of decay rates are feasible for different non-equilibrium systems over lengthy periods of time when it comes to diffusion. In engineering, biology, and ecology, the problems under study are used to represent spatial impacts. The fast Adomian decomposition method (FADM) is used to solve time fractional reaction diffusion equations, which are models of physical phenomena, in the current study. Caputo fractional derivative meaning applies to the specified time derivative. The results are in series form and correspond to the proposed fractional order problem. These models have a strong physical foundation, and their numerical treatments have significant theoretical and practical applications. The leaning of the rapid convergence of method-formulated sequences towards the appropriate solution is also graphically depicted. With less computational cost, this solution quickly converged to the exact solution. [ABSTRACT FROM AUTHOR]

The main purpose of this research is to inquire the new solitary wave solution of the coupled time-fractional models to validate the influence and proficiency of the planned variational iteration method (VIM) using conformable derivative definition. Applications to four demanding nonlinear problems like Hirota-Satsuma coupled KdV equations, modified Boussinesq (MB) equation, approximate long wave (ALW) equation and Drinfeld-Sokolov-Wilson (DSW) equation demonstrate the efficiency and the robustness of the method. An analysis of the consequences with effects of relevant parameters and comparison with the exact solution presented with the help of graphs tables and gives the further understanding of numerical results by others. The convergence of the method is illustrated numerical and their physical significance is discussed. [ABSTRACT FROM AUTHOR]

In real world physical applications purpose, it is complicated to acquire an exact solution of fuzzy differential equations due to complexities in fuzzy arithmetic and therefore creating the need for the use of reliable and efficient techniques in the solution of fuzzy differential equations. The purpose of this research paper is to utilize the reliable analytic approach of homotopy perturbation Sumudu transform method for better understanding of systems of non-linear fuzzy integro-differential equations, while using the concept of fuzzy parameter in certain dynamical problems to remove the hurdles faced in numerical approach. These mathematical models are of great interest in engineering and physics. Some numerical examples are also given to demonstrate the efficiency and superiority of the method, followed by graphical representation of the comparison of exact and approximated solution by using Maple 2017. [ABSTRACT FROM AUTHOR]

In this work, a new emerging analytical techniques variational iteration method combine with Aboodh transform has been applied to find out the significant important analytical and convergent solution of some mathematical models of fractional order. These mathematical models are of great interest in engineering and physics. The derivative is in Caputo's sense. These analytical solutions are continuous that can be used to understand the physical phenomena without taking interpolation concept. The obtained solutions indicate the validity and great potential of Aboodh transform with the variational iteration method and show that the proposed method is a good scheme. Graphically, the movements of some solutions are presented at different values of fractional order. [ABSTRACT FROM AUTHOR]

In this work, four variants of time-fractional nonlinear Caputo-type Newell- Whitehead-Segel (NWS) problem are solved by fractional variational iteration method. This equation results as mathematical model of many important physical reaction-diffusion processes and pattern formation theory in diverse fields of chemical engineering, mechanical engineering, bio-engineering, biology and fluid mechanics, and the numerical treatments of these models are of great scientific significance and practical values. The received outcomes and plotted graphs express that the carried out technique is unpretentious, competent, and easily applicable for nonlinear fractional-order Partial differential equations (PDEs). [ABSTRACT FROM AUTHOR]

In the present work, some mathematical models of non-linear types for first order integro-differential equation and system of fractional order are analytically tackled by the fractional VIM. The fractional derivative is considered in Caputo Fabrizio sense involving an exponential function and has interesting properties. The illustrative numerical examples show that the solution approach to the classical solutions as the order of the fractional derivative increases. The graphs obtained by generalized results are also plotted by using symbolic software Maple. Comparison of our approximate solution with the exact numerical results of the problems is presented to illustrate the reliability and efficiency of the method. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show use of modified variational iteration technique for understanding fuzzy integro-differential equations. Illuminating integro differential equations with correct parameter inside numerous displaying physical issues isn't exactly simple or better to state unthinkable in genuine issues. To defeat this trouble fuzzy idea is utilized. Generally it's hard to comprehend nonlinear fuzzy IDEs numerically. [ABSTRACT FROM AUTHOR]

In this paper, we have given the procedure and implement the differential transform method (DTM) to solve some linear fractional differential equations under fuzzy initial value conditions. The derivative is in the sense of the Caputo fractional derivative. The solutions of considered model examples are computed in the form of convergent series with easily computable components. To display the strength and applicability of this proposed method few illustrative examples are solved and the obtained results show that the said approach is easy to implement. [ABSTRACT FROM AUTHOR]

This paper is the witness of coupling of recently developed Aboodh transform and differential transform method. This coupling is known as Aboodh differential transform method. The proposed method is tested on some linear and nonlinear problems for suitable analytical solution and the obtained results demonstrate reliability and efficiency of the proposed method. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation. This new method is more efficient and easy to handle such differential equations in comparison to other methods. The results reveal the complete reliability that this method is very efficient, simple and applications hence can be extended to other problems of diverse nature. [ABSTRACT FROM AUTHOR]

Many mathematical models are contributed to give rise to of linear and nonlinear integral equations. This paper deals with the application of coupling of Laplace transform with decomposition method to solve analytically various nonlinear singular integral equations that described heat transfer. The elegant coupling gives rise to the modified versions of decomposition method which is very efficient in solving nonlinear problems of diversified nature of handling the Abel-type singular equation. It is observed that the decomposition method using Laplace transform is very efficient, easier to implements and more users friendly. Several examples are given to reconfirm the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we apply a modified version of the variational iteration method (MVIM) for solving K-S equation. The proposed modification is made by introducing He's polynomials in the correction functional of the variational iteration method (VIM). The nonlinear term can be easily handled by the use of He's Polynomials. The use of Lagrange multiplier coupled with He's polynomials are the clear advantages of this technique over the decomposition method. [ABSTRACT FROM AUTHOR]

In this paper we shall consider the various types of linear and non-linear fuzzy integro-differential equations of second kind like fuzzy Volterra integro-differential equation, fuzzy Fredholm integro-differential equation and mixed fuzzy Volterra-Fredholm integro-differential equation and suggested the algorithm of advanced numerical technique like Laplace homotopy perturbation method to find out their solutions. The application of this method on various types of fuzzy integro-differential equations shows the efficiency and reliability of the proposed method. [ABSTRACT FROM AUTHOR]

In this study, a reliable Homotopy Analysis Method (HAM) is applied to obtain analytical solutions of dielectric relaxation processes equation of fractional order. The, fractional derivatives are based on Jumarie's fractional derivative sense. The obtained solutions which are valid at low temperatures are applied to the dielectric relaxation processes. The graphical representation of the obtained results is courageous. [ABSTRACT FROM AUTHOR]

In this article one of the most reliable and effective method, (G' /G, 1/G) - expansion method has been employed to obtain exact traveling wave solutions of highly nonlinear partial differential equations (PDEs). The set of abundant exact traveling wave solutions of two very important nonlinear evolution equations of mathematical physics, i.e., modified Benjamin-Bona-Mahony (mBBM) and (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equations are developed. The comparison of the obtained numerical results with the existing along with the graphical representation is presented. It is shown that the Bi variable (G' /G, 1/G) -expansion method is a potent and very concise mathematical technique for solving nonlinear problems. [ABSTRACT FROM AUTHOR]

In this paper, a new mathematical technique Differential Transform Method (DTM) is presented for solving variants of nonlinear Klein-Gordon equation. The Differential Transform Method (DTM) provides an effective tool to solve analytically partial differential equations. The method is demonstrated by several examples. The obtained results indicate that this method to be very effective and simple. The method can easily be applied to many problems and is capable of reducing the size of computational work. [ABSTRACT FROM AUTHOR]

In this study we present the analysis of Adomian Decomposition Method using He's Polynomials for nonlinear WhithamBroer-Kaup equations dealing with propagation of shallow water waves with different dispersion relations. The exact solutions of two variants of WhithamBroer-Kaup equations are studied. The suggested method is used without discretization, linearization or restrictive assumptions. Numerical results show that the proposed method was efficient and capable to obtain the exact solution of this set of wave equations. The obtained solutions of these equations could straightforwardly show some facts of the described process deeply such as the propagation. It is clear that this method can be easily extended to other nonlinear wave equations arising in mathematical physics. [ABSTRACT FROM AUTHOR]

In this paper, Laplace decomposition method using He's polynomial is employed to determine the exact solution of the Burger equation which 1-dimensional nonlinear partial differential equation. This method is combined form of the Laplace Transforms Adomian decomposition method .The nonlinear term can be easily handled by the use of He's Polynomials. The explicit solutions obtained were compared the exact solutions. The method finds the solution without any restrictive assumptions and free from round-off errors and therefore reduce the numerical computation to a great extent. The method is tested on two examples and results show that new method is more effective and convenient to use and high accuracy of it is evident. [ABSTRACT FROM AUTHOR]

In this letter, we develop analytic solutions for integral equations within local fractional derivative operators by using the proposed local fractional Adomain's decomposition method. The iteration procedure is based on local fractional derivative. Two examples are given to elucidate the solution procedure, and the results are compared with the obtained results. The results reveal that the methodology is very efficient and simple tool for solving fractal phenomena arising in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]