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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.

This study introduces new operational matrices to approximate the solution of first and second order delay differential equations (DDEs) using Said-Ball polynomials (SBP). To obtain the required ap...

In this paper, a Cournot oligopoly with isoelastic demand function and constant marginal cost is considered. The local stability conditions of the Cournot equilibrium are determined for four models with different decision mechanisms. In the first model, firms adjust their outputs using the best reply response with naive expectations. The second model is a generalization of the first one, where firms have adaptive expectations. Meanwhile, the third and fourth models adopt the bounded rationality and local monopolistic approximation, respectively. The results show that, in the case of identical firms, the Cournot equilibrium is always stable when the firms adopt the local monopolistic approximation mechanism.

This research focuses on the approximate solutions of second-order fuzzy differential equations with fuzzy initial condition with two different methods depending on the properties of the fuzzy set theory. The methods in this research based on the Optimum homotopy asymptotic method (OHAM) and homotopy analysis method (HAM) are used implemented and analyzed to obtain the approximate solution of second-order nonlinear fuzzy differential equation. The concept of topology homotopy is used in both methods to produce a convergent series solution for the propped problem. Nevertheless, in contrast to other destructive approaches, these methods do not rely upon tiny or large parameters. This way we can easily monitor the convergence of approximation series. Furthermore, these techniques do not require any discretization and linearization relative with numerical methods and thus decrease calculations more that can solve high order problems without reducing it into a first-order system of equations. The obtained results of the proposed problem are presented, followed by a comparative study of the two implemented methods. The use of the methods investigated and the validity and applicability of the methods in the fuzzy domain are illustrated by a numerical example. Finally, the convergence and accuracy of the proposed methods of the provided example are presented through the error estimates between the exact solutions displayed in the form of tables and figures.

Delay differential equations (known as DDEs) are a broad use of many scientific researches and engineering applications. They come because the pace of the shift in their mathematical models relies all the basis not just on their present condition, but also on a certain past cases. In this work, we propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using the Homotopy Perturbation Method with double parametric form fuzzy numbers. The detailed algorithm of the approach to fuzzification and defuzzificationis analysis is provided. In the initial conditions of the proposed problem there are uncertainties with regard to the triangular fuzzy number. A double parametric form of fuzzy numbers is defined and applied for the first time in this topic for the present analysis. This method's simplicity and ability to overcome delay differential equations without complicating Adomian polynomials or incorrect nonlinear assumptions. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show the features of this proposed method, a numerical example is illustrated, involving first order fuzzy delay differential equation. These findings indicate that the suggested approach is very successful and simple to implement.

In this paper, we studied the 2D steady laminar boundary layer flow and heat transfer of Casson based nanofluid over an exponentially vertical stretching and shrinking sheet using one phase model. The thermal radiation and heat source/sink parameters are incorporated in the heat transfer equation and the slip parameters for the velocity and temperature are considered in the boundary conditions. The similarity variables have been used to convert the governing equations as a system of partial differential equations to the ordinary differential equations. The transformed equations are then solved by applying shooting technique, shootlib in Maple software. The numerical solutions for the governing equations indicate that the triple solutions arise when high suction is imposed for both stretching/shrinking sheet at certain ranges of the pertinent parameters. To examine the stability of the solutions, stability analyses is done by using BVP4c in Matlab software. The first solution is found to be a stable solution and physical realizable while the remaining solutions are not stable. The inclusion of three different nanoparticles shows that Copper – Casson nanofluid has the highest heat transfer rate compared to Aluminum-Casson and Graphite Oxide-Casson nanofluid.

From the observed datasets, we should be able to produce curve surfaces that have the same characteristics as the original datasets. For instance, if the given data are positive, then the resulting curve or surface must be positive on entire given intervals, i.e., everywhere. In this study, a new partial blended rational bi-quartic spline with C1 continuity is constructed through the partially blended scheme. This rational spline is defined on four corners of the rectangular meshes. The sufficient condition for the positivity of rational bi-quartic spline is derived on four boundary curve networks. There are eight free parameters that can be used for shape modification. The first-order partial derivatives are estimated by using numerical techniques. We also show that the proposed scheme is local quadratic reproducing such that it can exactly reproduce the quadratic surface. We test the proposed scheme to interpolate various types of positive surface data. Based on statistical indicators such as the root mean square error (RMSE) and coefficient of determination (R2), we found that the proposed scheme is on par with some established schemes. In fact, it requires less CPU times (in seconds) to generate the interpolating surface on rectangular meshes. Furthermore, by combining the statistical indicators’ result and graphically visualizing the test functions, the proposed method has the capability to reconstruct very comparable smoothing interpolating positive surfaces compared to some existing schemes. This finding is significant in producing a better interpolating surface for computer graphics applications since the proposed scheme has a smaller error compared with existing schemes.

In this study, a new scheme for positivity preserving interpolation is proposed by using C1 rational quartic spline of (quartic/quadratic) with three parameters. The sufficient condition for the positivity rational quartic interpolant is derived on one parameter meanwhile the other two are free parameters for shape modification. These conditions will guarantee to provide positive interpolating curve everywhere. We tested the proposed positive preserving scheme with four positive data and compared the results with other established schemes. Based on the graphical and numerical results, we found that the proposed scheme is better than existing schemes, since it has extra free parameter to control the positive interpolating curve.

Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic B&eacute, zier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness, that is, in order to achieve C 1 continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact B&eacute, zier ordinates, and the quartic scheme is global and not local. Therefore, the quartic B&eacute, zier triangular has received less attention. In this work, we use Zhu and Han&rsquo, s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic B&eacute, zier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation, (b) assign the first derivative at the respective data, (c) form a triangular surface via convex combination from three local schemes with C 1 continuity along all adjacent triangles, and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R2) and smaller maximum error (Max Error), requires about 12.5% of the CPU time of the quartic B&eacute, zier triangular, and is on par with Shepard triangular-based schemes. Therefore, the proposed scheme is significant for use in visualizing large and irregular scattered data sets. Finally, we tested the proposed positivity-preserving interpolation scheme to visualize coronavirus disease 2019 (COVID-19) cases in Malaysia.

In this paper, a numerical procedure called multistage optimal homotopy asymptotic method (MOHAM) is introduced to solve multi-pantograph equations with time delay. It was shown that the MOHAM algorithm rapidly provides accurate convergent approximate solutions of the exact solution using only one term. A comparative study between the proposed method, the homotopy perturbation method (HPM) and the Taylor matrix method are presented. The obtained results revealed that the method is of higher accuracy, effective and easy to use.

In this paper we prove the existence and uniqueness of coincident (fixed) points for nonlinear mappings of any number of arguments under a (ψ, θ, φ)-weak contraction condition without O-compatibility. The obtained results extend, improve and generalize some well-known results in the literature to be discussed below. Moreover, we present an example to show the efficiency of our results.

In this paper we prove the existence of ϒ-fixed point for a multidimensional nonlinear mappings F : Xk → X defined on the partially ordered metric spaces and satisfying (ψ, θ, ϕ)-weak contractive conditions. Moreover, we prove the uniqueness of that fixed point under extra conditions to (ψ, θ, ϕ)-weak contractive conditions.

In this paper, we discuss the numerical solution of second-order nonlinear two-point fuzzy boundary value problems (TPFBVP) by combining the finite difference method with Newton’s method. Numerical example using the well-known nonlinear TPFBVP is presented to show the capability of the new method in this regard and the results are satisfied the convex triangular fuzzy number. We also compare the numerical results with the exact solution, and it shows that the proposed method is good approximation for the analytic solution of the given TPFBVP.

This paper discusses scattered data interpolation by using cubic Timmer triangular patches. In order to achieve C1 continuity everywhere, we impose a rational corrected scheme that results from convex combination between three local schemes. The final interpolant has the form quintic numerator and quadratic denominator. We test the scheme by considering the established dataset as well as visualizing the rainfall data and digital elevation in Malaysia. We compare the performance between the proposed scheme and some well-known schemes. Numerical and graphical results are presented by using Mathematica and MATLAB. From all numerical results, the proposed scheme is better in terms of smaller root mean square error (RMSE) and higher coefficient of determination (R2). The higher R2 value indicates that the proposed scheme can reconstruct the surface with excellent fit that is in line with the standard set by Renka and Brown&rsquo, s validation.

In this paper, an approximate analytical solution for solving the fuzzy Bratu equation based on variation iteration method (VIM) is analyzed and modified without needed of any discretization by taking the benefits of fuzzy set theory. VIM is applied directly, without being reduced to a first order system, to obtain an approximate solution of the uncertain Bratu equation. An example in this regard have been solved to show the capacity and convenience of VIM.

A prescribed boundary base on elliptic partial differential operators of method is presented in this study to generate Wang-Ball surfaces. Particularly, the study is heavily concerned with the biharmonic Wang-Ball surfaces, which enables the overall surface to be controlled and generated on boundary curves instead of utilizing set of control points. To show the proposed study method, Biharmonic Wang-Ball patches of degree 3 was chosen for the study. Keywords: Wang-Ball plynomial, biharmonic, differential equa-tinn.

In this study, a new cubic Timmer triangular patch is constructed by extending the univariate cubic Timmer basis functions. The best scheme that lies towards the control polygon is cubic Timmer curve and surface compared to the other methods. From the best of our knowledge, nobody has extended the univariate cubic Timmer basis to the bivariate triangular patch. The construction of the proposed cubic Timmer triangular patch is based on the main idea of the cubic Ball and cubic Bezier triangular patches construction. Some properties of the new cubic Timmer triangular patch are investigated. Furthermore, the composite cubic Timmer triangular patches with parametric continuity (C1) and geometric continuity (G1) are discussed. Simple error analysis between the triangular patches and one test function is provided for each continuity type. Numerical and graphical results are presented by using Mathematica and MATLAB.

A mathematical analysis is performed to study the flow and heat transfer phenomena of Casson based nanofluid with effects of the porosity parameter and viscous dissipation over the exponentially permeable stretching and shrinking surface. The considered nanofluid comprises Casson as a base fluid that contains silver (Ag) and copper (Cu) solid nanoparticles. The system of the nonlinear governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) by applying similarity transformation. The obtained ODEs are solved by using shooting technique in Maple software. Numerically obtained results reveal dual solutions for various values of pertinent parameters. Due to occurrence of dual solutions, the stability analysis is done in order to find stable solution. Positive signs of smallest eigenvalues point out that the first solution is stable and second unstable. The variation of the velocity and the temperature profiles with coefficient of the skin friction and the Nusselt number are shown graphically. Both temperature profiles and its boundary layer thicknesses increase as volume fraction of nanoparticles of Ag and Cu are increased in the Casson fluid. Velocity profiles and corresponding boundary layer thicknesses decrease by suspension of nanoparticles of silver and copper, whereas the silver Ag nanoparticles show the greater rate of heat transfer enhancement as compared to copper Cu nanoparticles when suspended in Casson fluid., Mechanical engineering; Casson nanofluid; Shooting method; Dual solution; Stability analysis; Stretching/shrinking surface.

Let f be a circle homeomorphism with countably many break points that is, differentiable except in countably many points where the derivatives have a jump. Assuming its rotation number ρ to be irrational, we provide a necessary condition for the absolute continuity of invariant measure with respect to the Lebesgue measure.

This article proves that the robust finite-time synchronization behavior can be achieved for the chaotic systems with different orders. Based on the Lyapunov stability theory and using the nonlinear feedback control, sufficient algebraic conditions are derived to compute the linear controller gains. These gains are then used to achieve the robust finite time increasing order and reduced order synchronization of the chaotic systems. This study also discusses the design of a controller that accomplishes the finite time synchronization of two chaotic systems of different dimensions under the effect of both unknown model uncertainties and external disturbances. Numerical simulation results are furnished to validate the theoretical findings.

The radiation data is important for solar data analysis which is significantly for modeling, designing on energy conversion system development and outputting it as a variable that is needed by other related scientific researches.Obtaining the complete data for solar intensity is a tough job when the technical problem is raised either from the sensor’s devices or faulty data transmission. Next, missing values occur accordingly.As a solution, missing value estimation has been introduced to substitute the absent values with imputed values.This article is highlighted to propose cubic Bezier interpolation as potential techniques to generate replacement values. Afterwards, compare the performance capability of few submitted interpolation techniques dealing with difference amount of simulated missing data. Based on our knowledge, curiosity works for replaceable values by applying Bezier interpolation on solar radiation data is limited. The cubic Bezier interpolation performs very well on both areas but the most effective condition is at the lower missing values and higher absent values that come from Alor Setar and Penang cities, respectively, which offer the lowest RMSE value and the r² value nearest to 1

In this article, a novel robust nonlinear controller approach is focused to study theoretically, the reduced-order synchronization phenomena of two unrelated time-delayed chaotic systems under the determined and unknown parameters. It is assumed that the two systems are perturbed by the bounded unstructured uncertainties and unknown external disturbance. Based on the Lyapunov–Krasovskii functional theory, a robust nonlinear synchronization controller is focused and a suitable Lyapunov functional is constructed so that they establish the globally asymptotical stability of the closed-loop at the origin. Subsequently, suitable adaptive laws of unknown parameters are designed to identify the unknown parameters. Finally, the effectiveness of the proposed reduced-order synchronization approach is verified by numerical simulations. A brief comparison of the present study with prior works has been given.

Active control strategy is a powerful control technique in synchronizing chaotic/hyperchaotic systems. Until now, active control techniques have been employed to synchronize chaotic systems with the same orders. The present study overcomes the limitations of synchronization of chaotic systems of similar dimensions using active control. In this article, the authors investigate the synchronization problem for a drive-response chaotic system with different orders under the effect of both unknown model uncertainties and external disturbance. Based on the Lyapunov stability theory and Routh–Hurwitz criterion, a robust generalized active control approach is proposed and sufficient algebraic conditions are derived to compute a suitable linear controller gain matrix that guarantees the globally exponentially stable synchronization. Two examples are presented to illustrate the main results, namely reduced-order synchronization between the hyperchaotic Lu and the unified chaotic systems and the increased-order synchronization between the unified chaotic and the hyperchaotic Lu systems. There are three main contributions of the present study: (a) generalization of the active control for synchronization of a class of chaotic systems with different orders; (b) a recursive approach is proposed to compute a suitable linear controller gain matrix and (c) reduced (increased) order synchronization under the effect of both unknown model uncertainties and external disturbances. A comparative study has been done with our results and previously published work in terms of synchronization speed and quality. Future applications of the proposed reduced (increased) order synchronization approach are discussed. Finally, numerical simulations are given to verify the effectiveness of the proposed reduced (increased) order active synchronization approach.

Complete solar radiation data is important in variety of disciplines such as solar radiation development system or solar project which is strongly affected by hourly variation of the solar radiation. However, the available data may contain missing data for several days in the absence of measurement due to several factors such as instrument failure and human errors. In this study, the secondary data of solar radiation at Penang, Malaysia taken from Malaysian Meteorology Department was investigated. In order to fill the gap of the above missing values, the interpolation method based on positivity preserving Bezier curves to ensure that the solar radiation are always positive is proposed. This paper start with the derivation of positivity preserving condition for Bezier curves by imposing lower bound of the Bezier control points. The proposed method is tested using simulated missing value of solar radiation dataset. Result from RMSE and Coefficient of Determination (r2) for different set of missing value from 10, 30 and 50%, show that the proposed method produce a better result compared to existing interpolation method without implementing positivity preserving condition. Finally, the proposed method was used to fill the gap of missing values of solar radiation at Penang, Malaysia.

In this work, we developed an approximate analytical method based on the optimal homotopy asymptotic method (OHAM) to solve fuzzy partial differential equations (FPDE). The method has been applied to fuzzy reaction-diffusion equation with initial condition. By means of illustrative examples, we demonstrated the accuracy, efficiency, and flexibility of the proposed method.

The purpose of this analysis would be to provide a computational technique for the numerical solution of second-order nonlinear fuzzy initial value (FIVPs). The idea is based on the reformulation of the fifth order Runge Kutta with six stages (RK56) from crisp domain to the fuzzy domain by using the definitions and properties of fuzzy set theory to be suitable to solve second order nonlinear FIVP numerically. It is shown that the second order nonlinear FIVP can be solved by RK56 by reducing the original nonlinear equation intoa system of couple first order nonlinear FIVP. The findings indicate that the technique is very efficient and simple to implement and satisfy the Fuzzy solution properties. The method’s potential is demonstrated by solving nonlinear second-order FIVP.

This book is a valuable resource for those engaged with mathematical modeling. The six chapters of this book discuss the recent trends in applied and associated mathematical sciences, focusing on techniques and modeling, based on real problems. With the appropriate mathematical models and interpretation of numerical results, this book provides useful information and guidance to understand real problems. This book will be useful for new and young researchers from different disciplines to link mathematics to real world applications.

A mathematical analysis is performed to study the flow and heat transfer phenomena of Casson based nanofluid with effects of the porosity parameter and viscous dissipation over the exponentially permeable stretching and shrinking surface. The considered nanofluid comprises Casson as a base fluid that contains silver ( A g ) and copper ( C u ) solid nanoparticles. The system of the nonlinear governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) by applying similarity transformation. The obtained ODEs are solved by using shooting technique in Maple software. Numerically obtained results reveal dual solutions for various values of pertinent parameters. Due to occurrence of dual solutions, the stability analysis is done in order to find stable solution. Positive signs of smallest eigenvalues point out that the first solution is stable and second unstable. The variation of the velocity and the temperature profiles with coefficient of the skin friction and the Nusselt number are shown graphically. Both temperature profiles and its boundary layer thicknesses increase as volume fraction of nanoparticles of A g and C u are increased in the Casson fluid. Velocity profiles and corresponding boundary layer thicknesses decrease by suspension of nanoparticles of silver and copper, whereas the silver A g nanoparticles show the greater rate of heat transfer enhancement as compared to copper C u nanoparticles when suspended in Casson fluid.