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Abstract A residual power series scheme is developed for mixed-type p-system of conservation laws. The residual power series method (RPSM) does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive computations. Convergence hypotheses are discussed and error bounds of exponential rates are derived. The scheme is numerically tested to several cases of the van der Waals equations which arise in fluid dynamics. Stability is obtained graphically while transition through regions. The reported results reveal the reliability, efficiency and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics. Highlights • A residual power series scheme is developed for mixed-type systems of conservation laws. • Convergence hypotheses are discussed and error bounds of exponential rates are derived. • The scheme is numerically tested on the van der Waals gas model with Riemann initial datum. • Stability is obtained graphically while transition through regions for several cases. • Reported results reveal the efficiency of the RPSM for handling NPDEs in applied mathematics. [ABSTRACT FROM AUTHOR]

In this research, the new auxiliary equation method (NAEM) for higher order nonlinear fractional Huxley equation is being employed to extricate the novel soliton solutions using Beta and M -Truncated fractional derivatives. For waves of finite amplitude, the Huxley equation demonstrates a substantial transfer of spectrum energy. A comparison of the solutions of the model with both fractional derivatives is also included in this research. Various kinds of solitary traveling wave solutions, such as trigonometric, hyperbolic, exponential, rational functions, etc., are found. These types of solutions demonstrate the superiority of the novelty of the method. This method's key advantage over others is that it provides more broad solutions with certain flexible parameters. 3D and 2D graphs are used graphically to demonstrate the dynamical structures of the solutions. The results are presented in a way that demonstrate the usefulness and competence of the approach used to handle various nonlinear fractional partial differential equations. Lastly, we investigate the comparison of the gain spectra for modulation instability and the depiction of certain noteworthy outcomes by illustratively depicting the 2 D figures produced by carefully considering the parameters • The solitons solutions of fractional Huxley equation is find out using new auxiliary equation method. • The behavior of the solutions is analyzed using fractional derivatives. • A comparative analysis is done on the behalf of fractional derivatives. • Modulation instability analysis is done of the fractional model. [ABSTRACT FROM AUTHOR]

The aim of the present analysis is to apply two relatively recent methods, reduced differential transform method (RDTM) and differential transform method (DTM), for the solution of balance law systems. New generalized transformed formulas are derived. The new approaches provided the solution in the form of a rapidly convergent series with easily computable components in the RDTM case, and costly components for the DTM. A comparison between the two methodologies shows that the RDTM is more effective, efficient, powerful and able to be applicable for large class of nonlinear partial differential equations than the DTM. Two test modeling problems are discussed to illustrate the effectiveness and performance of RDTM. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we apply Adomian decomposition method (ADM) to develop a fast and accurate algorithm for systems of conservation laws of mixed hyperbolic elliptic type. The solutions of our model equations are calculated in the form of convergent power series with easily computable components. The results obtained are compared with our Modification of Adomian decomposition method (MADM) Az-Zo’bi and Al-Khaled (2010) [1]. The study outlines the significant features of the two methods. With application to van der Waals system, we obtain the stability of the approximate solutions graphically when the system changes type with more efficiency of the MADM. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new convergence proof of the Adomian’s decomposition method (ADM), applied to the generalized nonlinear system of partial differential equations (PDE’s) based on new formula for Adomian polynomials. The decomposition scheme obtained from the ADM yields an analytical solution in the form of a rapidly convergent series for a system of conservation laws. Systems of conservation laws is presented, we obtain the stability of the approximate solution when the system changes type. We show with an explicit example that the latter property is true for general Cauchy problem satisfying convergence hypothesis. The results indicate that the ADM is effective and promising. [ABSTRACT FROM AUTHOR]