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Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.

Incorporation of momentum gradients produced due to inertial motion of the lid along with the presence of temperature differences in the configuration make the physical problem more significant. The joint variation of momentum and thermal diffusion in diversified natural liquids is recognized as mixed convection. Valuable attention has been received by such a phenomenon in different areas of science and technology such as in wind current --based solar receivers, electronic instruments, control of emergency shutdown in reactors, thermal exchangers, oceanic currents, control of atmospheric pollution, and so on. So, the main focus is to contemplate hydrothermal characteristics of a power-law fluid contained in a square cavity with themovement of the upper lid and being thermally adiabatic. The other extremities are considered to be at rest, and the base wall is prescribed with uniform/nonuniform temperature distributions. The governing formulation of the problem is handled by executing a finite element approach. Hybrid meshing is performed for domain discretization, and weak variational formulation is utilized for formulation discretization. Second-degree polynomials are employed as the interpolation function, providing information about velocity and temperature distributions at boundary and intermediate nodes. The system of finalized non-linear equations is resolved by using the Paradiso software. The results for velocity and temperature distributions are attained comparatively for uniformly and non-uniformly heated profiles. The kinetic energy and average Nusselt number are also computed against flow concerning variables. From the attained graphical and tabular data, it is deduced that by increasing the Reynolds number, inertial forces dominate over buoyancy forces and the effect of lid movement is prominent on flow characteristics. It is also inferred that for the shear thickening case and for all values of the Reynolds number, the average Nusselt number shows a constant behavior. [ABSTRACT FROM AUTHOR]

Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an integral operator with a singular kernel. Second, using these relations to solve a mixed integral equation (MIE) of the second kind in the space L2[-1,1] x C[0, T], T < 1. The way to do this is to derive a general principal theorem of the spectral relations from the term of the Volterra-Fredholm integral equation (V-FIE), with the help of the Chebyshev polynomials (CPs), and then use the results in the general MIE to discuss its solution. More than that, some special and important cases will be devised that help explain many phenomena in the basic sciences in general. Here, the FI term is considered in position, in L2[-1,1], and its kernel takes a logarithmic form multiplied by a general continuous function. While the VI term in time, in C[0, T], T < 1, and its kernels are smooth functions. Many numerical results are considered, and the estimated error is also established using Maple 2022. [ABSTRACT FROM AUTHOR]

Enhanced heat transport in advanced nanofluids (ternary hybrid nanofluids) is one of the major demands of the time and is potentially contributing in food processing to maintain the temperature of building, cooling of electronic devices, paint industries and biomedical engineering. Therefore, an efficient heat transport model is developed in this study and innovative ternary mixture [(Al2O3-CuO-Ag)] along with feasible thermophysical attributes comprising the effects of ternary nanoparticles and similarity equations are exercised to obtain the desired sort of nanoliquid model. This model is related to vertically oriented cylinder with novel upgradation of permeability, upthrust forces and nonlinear solar thermal radiations. In the next stage, mathematical treatment of [(Al2O3-CuO-Ag)/H2O]thnf is done and successfully achieved the desired convergence and then organized the graphical results. The furnished results disclosed that tri-composites-based nanofluid has low velocity than hybrid and common nanofluids. Moreover, temperature in [(Al2O3-CuO-Ag)/H2O]thnf is dominant over both hybrid and mono nanofluids. The integrated effects of nonlinear thermal radiations are of much interest in the temperature enhancement and observed that Rd and θ w are better for thermal improvement. [ABSTRACT FROM AUTHOR]

In this research work, the two-dimensional (2D), incompressible fluid flow has been taken into consideration. The flow is supposed to be steady and laminar. By considering the water-based nanoparticles of SWCNTs and MWCNTs in the presence of thermal radiation, the rate of heat transferring and entropy generation effects in a regenerative cooling system of a rocket engine are evaluated. The effects of the length and radius of the nanomaterials on the problem are also considered. Solutions for temperature, velocity profile, irreversibility (entropy generation) and the Bejan number are discussed graphically, and the effects of various significant factors are considered on these profiles. The modeled physical problems in current exploration are dependent upon governing laws which appear in terms of PDEs. These PDEs are reformed into a system of nonlinear ODEs. We used numerical scheme known as (RK-4) in combination with the shooting iteration technique to obtain the solutions to transformed fluid flow equations, because the resultant ODEs are extremely nonlinear and finding the exact solution is very difficult. It is investigated that the Eckert number, nanoparticles volume fraction and radiation parameter upsurge the thermal field as well as the irreversibility of the system. Furthermore, the dual behavior of nanoparticles volume fraction and viscosity parameter on velocity profile is observed. Bejan number shows increasing effects in response to nanoparticles volume fraction and radiation parameter, whereas a reverse impact of Bejan number is noticed for the rising values of Eckert number. [ABSTRACT FROM AUTHOR]

This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space L 2 (− 1 , 1) × C [ 0 , T ] , (T < 1). The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equation's solution are discussed using Banach's fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each x with y. [ABSTRACT FROM AUTHOR]

Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique for employing the Laplace transform LT in solving the nonlinear time-fractional reaction–diffusion model. The new approach is called the Laplace-residual power series method (L-RPSM), which imitates the residual power series method in determining the coefficients of the series solution. The proposed method is also adapted to find an approximate series solution that converges to the exact solution of the nonlinear time-fractional reaction–diffusion equations. In addition, the method has been applied to many examples, and the findings are found to be impressive. Further, the results indicate that the L-RPSM is effective, fast, and easy to reach the exact solution of the equations. Furthermore, several actual and approximate solutions are graphically represented to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the Volterra–Hammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors. [ABSTRACT FROM AUTHOR]

The present computational model is built to analyze the energy and mass transition rate through a copper and cobalt ferrite water-based hybrid nanofluid (hnf) flow caused by the fluctuating wavy spinning disk. Cobalt ferrite (CoFe2O4) and copper (Cu) nanoparticles (nps) are incredibly renowned in engineering and technological research due to their vast potential applications in nano/microscale structures, devices, materials, and systems related to micro- and nanotechnology. The flow mechanism has been formulated in the form of a nonlinear set of PDEs. That set of PDEs has been further reduced to the system of ODEs through resemblance replacements and computationally solved through the parametric continuation method. The outcomes are verified with the Matlab program bvp4c, for accuracy purposes. The statistical outputs and graphical evaluation of physical factors versus velocity, energy, and mass outlines are given through tables and figures. The configuration of a circulating disk affects the energy transformation and velocity distribution desirably. In comparison to a uniform interface, the uneven spinning surface augments energy communication by up to 15%. The addition of nanostructured materials (cobalt ferrite and copper) dramatically improves the solvent physiochemical characteristics. Furthermore, the upward and downward oscillation of the rotating disc also enhances the velocity and energy distribution. [ABSTRACT FROM AUTHOR]

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions. [ABSTRACT FROM AUTHOR]

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values. [ABSTRACT FROM AUTHOR]