This paper investigates fractional-order partial differential equations analytically by applying a modified technique called the Laplace residual power series method. The analytical solution was utilized to test the accuracy and precision of the proposed methodologies and shown by tables and graphs. The solution is a convergent series established on Taylor's new form. When determining the series coefficients like RPSM, the fractional derivatives must be calculated every time. We only need to perform a few computations to obtain the coefficients because LRPSM only requires the concept of an infinite limit. The advantage of this method is that it does not require Adomian polynomials or he's polynomials to solve nonlinear problems. As a result, the method's reduced computation size is a strength. The outcome we got supports the idea that the suggested method is the best one for handling any non-linear models that appear in technology and science.

In this article, a new generalized iterative technique is presented for finding the approximate solution of a system of linear equations $Ax=b$. The efficiency of iterative technique is analyzed by implementing it on some examples, and then comparing with existing methods. A parameter introduced in the method plays very vital role for a better and rapid solution. Convergence analysis is also examined. Findings of this paper may stimulate further research in this area.

The aim of this article is to present a comparison of two analytical approaches toward obtaining the solution of the time-fractional system of partial differential equations. The newly proposed approaches are the new approximate analytical approach (NAAA) and Mohand variational iteration transform approach (MVITA). The NAAA is based on the Caputo-Riemann operator and its basic properties with the decomposition procedure. The NAAA provides step wise series form solutions with fractional order, which quickly converge to the exact solution for integer order. The MVITA is based on a variational iteration procedure and uses the Mohand integral transform. The MVITA also provides a series solution without a stepwise solution. Both approaches provide a series form of solutions to the proposed problems. The analytical procedures and obtained results are compared for the proposed problems. The obtained results were also compared with exact solutions for the problems. The obtained result and plots have shown the validity and applicability of the proposed algorithms. Both approaches can be extended for the analytical solution of other physical phenomena in science and technology.

The present paper is the result of a contemplative study of a multi-time control problem (MCP) by considering its associated equivalent auxiliary control problem (MCP) ${}_{\varsigma }$ via the exact $l_1$ penalty method. Further study reveals that the solution set of the considered problem and the auxiliary problem exhibits an equivalence under the KT-pseudoinvexity hypothesis. Moreover, the study is extended towards the saddle point defined for (MCP) to establish the relationship between the solution set of multi-time control problem (MCP) and its associated equivalent auxiliary control problem (MCP) ${}_{\varsigma }$. Finally, we present an illustrative application to authenticate the results presented in this paper.

The two approaches to solving nonlinear Caputo time-fractional wave-like equations with variable coefficients are examined in this study. The Homotopy perturbation transform method and the Yang transform decomposition method are the names of these two techniques. Three separate numerical examples are provided to demonstrate the effectiveness and precision of the suggested methods. The results were acquired to demonstrate the effectiveness and power of the two approaches, providing estimates with better precision and closed form solutions. The solutions to these kinds of equations can be found using the suggested methods as infinite series, and when these series are in closed form, they provide the exact solution. The suggested techniques have been demonstrated to be effective and efficient in their application. Three numerical examples are used to examine the methods accuracy and effectiveness.

The role of integral inequalities can be seen in both applied and theoretical mathematics fields. According to the definition of convexity, it is possible to relate both concepts of convexity and integral inequality. Furthermore, convexity plays a key role in the topic of inclusions as a result of its definitional behavior. The importance and superior applications of convex functions are well known, particularly in the areas of integration, variational inequality, and optimization. In this paper, various types of inequalities are introduced using inclusion relations. The inclusion relation enables us firstly to derive some Hermite-Hadamard inequalities (H.H-inequalities) and then to present Jensen inequality for harmonical $h$-Godunova-Levin interval-valued functions (GL-IVFS) via Riemann integral operator. Moreover, the findings presented in this study have been verified with the use of useful examples that are not trivial.

In this paper, we investigate the semi-analytical solution of Kaup-Kupershmidt equations with the help of a modified method known as the new iteration transformation technique. This method combines the Yang transform and the new iteration technique. The nonlinear terms can be calculated straightforwardly by a new iteration method. The numerical simulation results have been presented to demonstrate the reliability and validity of the proposed approach. The result confirms that the suggested technique is the best tool for dealing with any nonlinear problems arising in technology and science. In addition, in terms of figures for varying fractional order, the physical behavior of new iteration transformation technique solutions has been shown and the numerical simulation is also exhibited. The solutions of the new iteration transformation technique reveal that the projected technique is reliable, competitive and powerful for studying complex nonlinear fractional type models.

In this paper, we study the forced oscillation of solutions of a fractional partial differential system with damping terms by using the Riemann-Liouville derivative and integral. We obtained some new oscillation results by using the integral averaging technique. The obtained results are illustrated by using some examples.

In this paper, we establish an integral equality involving a multiplicative differentiable function for the multiplicative integral. Then, we use the newly established equality to prove some new Simpson's and Newton's inequalities for multiplicative differentiable functions. Finally, we give some mathematical examples to show the validation of newly established inequalities.

In this study, we implemented the Yang residual power series (YRPS) methodology, a unique analytical treatment method, to estimate the solutions of a non-linear system of fractional partial differential equations. The RPS approach and the Yang transform are togethered in the YRPS method. The suggested approach to handle fractional systems is explained along with its application. With fewer calculations and greater accuracy, the limit idea is used to solve it in Yang space to produce the YRPS solution for the proposed systems. The benefit of the new method is that it requires less computation to get a power series form solution, whose coefficients should be established in a series of algebraic steps. Two attractive initial value problems were used to test the technique's applicability and performance. The behaviour of the approximative solutions is numerically and visually discussed, along with the effect of fraction order $\varsigma$. It was observed that the proposed method's approximations and exact solutions were completely in good agreement. The YRPS approach results highlight and show that the approach may be utilized to a variety of fractional models of physical processes easily and with analytical efficiency.