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In this work, the fuzzy fractional two-point boundary value problems (FFTBVPs) are analyzed and solved using the fuzzy fractional homotopy analysis method (FF-HAM). Fuzzy set theory mixed with Caputo fractional derivative properties is utilized to produce a new formulation of the standard HAM in the fuzzy domain for the persistence of approximation series solutions for fuzzy fractional differential equations with boundary conditions. The FF-HAM provides a suitable way of controlling the convergence of the series solution through the advantage of the convergence control parameter, which plays a pivotal role in solving a wide range of mathematical problems. The convergence analysis algorithm has been described, along with a graphical representation of the FF-HAM of the proposed applications. The method produces high accuracy solutions with simple implementation for solving linear and nonlinear fuzzy fractional boundary value problems associated with physical application. Also, the obtained results are analyzed and compared with those present in the literature to show the efficiency of the FF-HAM.

Hybrid nanofluid has an extensive range of real-world applications. Hybrid nanofluid is a new and advanced nanofluid modification extensively used to increase thermal efficiency in fluid flow systems. The main objective of this research is to study magnetohydrodynamics hybrid nanofluid flow numerically in two dimensional over a vertical exponentially shrinking sheet, considering the effects of Joule heating and thermal slip condition. Furthermore, using the Tiwari-Das model, the influence of the suction parameter on variations of reduced skin friction and reduced heat transfer is also explored. The hybrid nanofluid in this research is an Al2O3+Cu/water hybrid nanofluid, in which water is the base fluid, and two types of solid nanoparticles, namely Alumina (Al2O3) and copper (Cu), are combined together. The governing partial differential (PDEs) equations are transformed into the ordinary differential equations (ODEs) using exponential similarity variables. The resulting ordinary differential equations (ODEs) are numerically solved using the three-stage Labatto III-A technique in the "MATLAB software's" bvp4c solver. Hybrid nanofluids have greater thermal efficiency than nanofluids and base fluid. Dual solutions are obtained in specified ranges of suitable parameters. The temperature profile rises in both solutions as the Eckert value enhances. Besides, In the first and second solutions, the thermal boundary layer thickness decreased gradually as the thermal slip parameter increased. Finally, the conclusions presented that solution duality exists when the suction parameter , while no flow of fluid is possible when .

The numerical investigation of unsteady magnetohydrodynamic (MHD) stagnation point flow of Al2O3-Cu/H2O hybrid nanofluid past a convectively heated permeable stretching/shrinking sheet with suction/injection effect is underlined. The characteristic of MHD and boundary condition with suction/injection has received a lot of consideration due to its across-the-board application in mechanical and chemical engineering. The governing continuity, momentum and energy equations are transformed into a system of nonlinear ordinary differential equations using similarity transformation, which is then solved using the bvp4c routine. Numerical results are obtained for the skin friction coefficient, local Nusselt number as well as the velocity and temperature profiles for certain values of the governing parameters, namely suction/injection parameter, copper nanoparticle volume fraction parameter and MHD parameter. Results showed that both velocity profile and temperature increase as the suction/injection parameter increases for the first solution and decreases for the second solution. Similarly, when increase the value of MHD parameter M, the velocity and temperature profiles are decreases for both solutions. The magnitude of the reduced skin friction coefficient and the local Nusselt number are notably increased for the first solution with increasing values of the suction/injection and MHD parameters. Finding also revealed that the skin friction coefficient is intensified in conjunction with the local Nussel number by adding up the copper nanoparticle volume fraction. In general, dual solutions are found to exist to a particular extent of the stretching/shrinking sheet.

In this work, the Optimal Homotopy Asymptotic Method (OHAM) is prolifically implemented to find the optimal solutions of fractional order of fuzzy differential equations. We inspect the competence of the method by examining fractional order time-dependent first order fuzzy initial value problems in the Caputo derivative sense. The concepts of fuzzy sets theory and fractional calculus were used to present a new general fuzzification formulation of the OHAM method followed by fuzzy analysis for the proposed fractional problems are presented in detail. OHAM is recognized as a feasible and convergent method for solving linear and nonlinear models, and it provides a convenient technique for controlling the convergence of approximate solutions optimally. The method's capability is demonstrated by obtaining approximate solutions to linear and nonlinear fuzzy initial value problems in various orders of fractional derivative. The numerical results acknowledge that OHAM is a feasible and transformation mechanism for solving linear and nonlinear fuzzy fractional initial value problems. Also, the results ensure that the method used was succinct, efficient, and simple to use, to deal with a more comprehensive fuzzy fractional differential equation.

Hybrid nanofluid is considered a modern and improvised form of nanofluid which usually used to enhance the performance of heat transfer in fluid flow systems. Previous studies found hybrid nanofluid offered a wide range of applications and this opened up numerous new opportunities to further explore the unknown behaviour of hybrid nanofluid under different body geometries and physical parameters. This paper numerically studied a two-dimensional mixed convection and radiative Al2O3-Cu/H2O hybrid nanofluid flow over a vertical exponentially shrinking sheet with partial slip conditions. The main objective is to investigate the effect of mixed convection and radiation on the velocity and temperature profiles, as well as the effect of suction on reduced skin friction and reduced heat transfer with respect to solid volume fraction of copper, velocity, and thermal slips. Exponential similarity variables transformed the governing system of partial differential equations into a system of ordinary differential equations which is solved via MATLAB’s bvp4c solver. Outcomes showed that the value of the reduced heat transfer upsurges in the first solution but declines in the second solution when the velocity slip rises. The reduced heat transfer decreases in both dual solutions when thermal slip is enhanced. As the intensity of thermal slip increases, the reduced skin friction rises in the first solution and decreases in the second. As the mixed convection parameter increases, no obvious variation is noticed in the temperature distribution within the first solution, but increasing trend is observed within the second solution. An increment in the temperature distribution also observed within the dual solutions as the thermal radiation parameter increases. In summary, findings from this study are particularly useful to understand various behaviour of Al2O3-Cu/H2O hybrid nanofluid under the influence of mixed convection, radiation, and partial slip conditions when it flows over a vertical exponential shrinking sheet.

In this paper we prove a new version of Kransoselskii's fixed-point theorem under a ($\psi, \theta, \varphi$)-weak contraction condition. The theoretical result is applied to prove the existence of a solution of the following fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators orders of $00:$ \begin{equation}\nonumber \left\{\begin{array}{ll} D^{\alpha}[x(t)-f(t, x(t))]=g(t, x(t), I^{\beta}(x(t))), \,\,\, \text{a.e.} \,\,\, t\in J,\,\, \beta>0,\\ x(t_{0})=x_{0}, \end{array} \right. \end{equation} where $D^{\alpha}$ is the Riemann-Liouville fractional derivative order of $\alpha,$ $I^{\beta}$ is Riemann-Liouville fractional integral operator order of $\beta>0,$ $J=[t_{0}, t_{0}+a],$ for some fixed $t_{0}\in \mathbb{R},$ $a>0$ and the functions $f:J\times \mathbb{R}\rightarrow \mathbb{R}$ and $g:J\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ satisfy certain conditions. An example is also furnished to illustrate the hypotheses and the abstract result of this paper.

Let be one-parameter family of circle homeomorphisms with a break point, that is, the derivative has jump discontinuity at this point. Suppose satisfies a certain Zygmund condition which is dependent on parameter . We prove that the renormalizations of circle homeomorphisms from this family with rational rotation number of sufficiently large rank are approximated by piecewise fractional linear transformations in and -norms, depending on the values of the parameter and , respectively.

In this paper, we prove the existence of a solution of a fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators by utilizing a new version of Kransoselskii-Dhage type fixed-point theorem obtained in [13]. Moreover, we provide an example to support our result.

In this paper, we prove an existence of Υ-fixed point as multidimensional fixed point in partially ordered quasi metric space under (ψ, θ, ϕ)-weak contraction conditions. This result extends and generalizes some existing results in the literature.

Any first order initial value problem can be integrated numerically by discretizing the interval of integration into a number of subintervals, either with equally distributed grid points or non-equally distributed grid points. Hence, as the integration advances, the numerical solutions at the grid points are calculated and being known. However, the numerical solutions between the grid points remain unknown. This will form difficulty to individuals who wish to study a particular solution which may not fall on the grid points. Therefore, some sorts of interpolation techniques are needed to deal with such difficulty. Spline interpolation technique remains as a well known approach to approximate the numerical solution of a first order initial value problem, not only at the grid points but also everywhere between the grid points. In this short article, a new quartic spline method has been derived to obtain the numerical solution for first order initial value problem. The key idea of the derivation is to treat the third derivative of the quartic spline function to be a linear polynomial, and obtain the quartic spline function with undetermined coefficients after three integrations. The new quartic spline function is ready to be used when all unknown coefficients are found. We also described an algorithm for the new quartic spline method when used to obtain the numerical solution of any first order initial value problem. Two test problems have been used for numerical experimentations purposes. Numerical results seem to indicate that the new quartic spline method is reliable in solving first order initial value problem. We have compared the numerical results generated by the new quartic spline method with those obtained from an existing spline method. Both methods are found to have comparable accuracy.

Numerical methods that are based on rational functions or better known as rational methods were discovered 60 years ago when they were initially used to deal with problem whose solution possess singularity. Ever since then, a number of studies have discovered various types of rational methods and used them to solve more general first order initial value problems such as stiff problem and problem with oscillatory property. Previous studies showed the reliability of rational methods in solving first order initial value problem through numerical experimentations. In this article, we have investigated the solvability of several existing fourth order rational methods to second order boundary value problem by replacing it with a coupled of first order initial value problems. Such replacement yielded the shooting method. As part of the investigation, these rational methods were compared among themselves and also compared with the 4-stage fourth order explicit Runge-Kutta method. Numerical experimentations seemed to indicate that rational methods were more accurate than the Runge-Kutta method for the case of nonlinear boundary value problem but vice versa for the case of linear boundary value problem.

In this study, the concept of block multistep methods based on rational approximants is introduced for the numerical solution of first order initial value problems. These numerical methods are also called rational block multistep methods. The main reason to consider block multistep methods in rational setting, is to improve the numerical accuracy and absolute stability property of existing block multistep methods that are based on polynomial approximants. For this pilot study, a 2-point explicit rational block multistep method is developed. Local truncation error and stability analysis for this new method are included as well. Numerical experimentations and results using some test problems are presented. Numerical results are satisfying in terms of numerical accuracy. Finally, future issues on the developments of rational block multistep methods are discussed.

An explicit two-step, second order rational method for the numerical solution of first order initial value problems is introduced in this paper. Existing rational multistep methods required the computations of higher derivatives from a given initial value problem. However, the new two-step rational method does not require any computation of these higher derivatives, and thus save up some computational cost. Numerical results showed that the new rational multistep method and existing rational multistep method are found to have comparable accuracy in solving first order initial value problems.

In this paper, a new implicit Runge-Kutta method which based on a 4-point Gauss-Kronrod-Radau II quadrature formula is developed. The resulting implicit method is a 4-stage sixth order Gauss-Kronrod-Radau IIA method, or in brief as GKRM(4,6)-IIA. GKRM(4,6)-IIA requires four function of evaluations at each integration step and it gives accuracy of order six. In addition, GKRM(4,6)-IIA has stage order four and being L-stable. Numerical experiments compare the accuracy between GKRM(4,6)-IIA and the classical 3-stage sixth order Gauss-Legendre method in solving some test problems. Numerical results reveal that GKRM(4,6)-IIA is more accurate than the 3-stage sixth order Gauss-Legendre method because GKRM(4,6)-IIA has higher stage order.