Difference Equations

In this study, we use a compartmental nonlinear deterministic mathematical model to investigate the effect of different optimal control strategies in controlling Tuberculosis (TB) disease transmission in the community. We employ stability theory of differential equations to investigate the qualitative behavior of the model by obtaining the basic reproduction number and determining the local stability conditions for the disease-free and endemic equilibria. We consider three control strategies representing distancing, case finding, and treatment efforts and numerically compare the levels of exposed and infectious populations with and without control strategies. The results suggest that combination of all controls is the best strategy to eradicate TB disease from the community at an optimal level with minimum cost of interventions.

The nutrition of pregnant women is crucial for giving birth to a healthy baby and even for the health status of a nursing mother. In this paper, the poor nutrition in the life cycle of humans is explored in the fractional sense. The proposed model is examined via the Caputo fractional operator and a new one with Mittag-Leffler (ML) nonsingular kernel. The stability analysis as well as the existence and uniqueness of the solution are investigated, and an efficient numerical scheme is also designed for the approximate solution. Comparative numerical analysis of these two operators reveals that the model based on the new fractional derivative with ML kernel has a different asymptotic behavior to the classic Caputo. Thus, the new aspects of fractional calculus provide more flexible models which help us to adjust the dynamical behaviors of the real-world phenomena better.

In this paper we study the concept of analyticity for complex-valued functions of a complex time scale variable, derive a time scale counterpart of the classical Cauchy-Riemann equations, introduce complex line delta and nabla integrals along time scales curves, and obtain a time scale version of the classical Cauchy integral theorem.

The present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number R 0 $\mathcal{R}_{0}$ is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kut...

Fractal analysis is one of interesting research areas of computer science and engineering, which depicts a precise description of phenomena in modeling. Visual beauty and self-similarity has made it an attractive field of research. The fractal sets are the effective tools to describe the accuracy of the inequalities for convex functions. In this paper, we employ linear fractals R α to investigate the (s, m)-convexity and relate them to derive generalized Hermite-Hadamard (HH) type inequalities and several other associated variants depending on an auxiliary result. Under this novel approach, we aim at establishing an analog with the help of local fractional integration. Meanwhile, we establish generalized Simpson-type inequalities for (s, m)-convex functions. The results in the frame of local fractional showed that among all comparisons, we can only see the correlation between novel strategies and the earlier consequences in generalized s-convex, generalized m-convex, and generalized convex functions. We obtain application in probability density functions and generalized special means to confirm the relevance and computational effectiveness of the considered method. Similar results in this dynamic field can also be widely applied to other types of fractals and explored similarly to what has been done in this paper.

We investigate the existence of solutions for an increasing variables singular m-dimensional system of fractional q-differential equations on a time scale. In this singular system, the first equation has two variables and the number of variables increases permanently. By using some fixed point results, we study the singular system under some different conditions. Also, we provide two examples involving practical algorithms, numerical tables, and some figures to illustrate our main results. MSC: Primary 34A08; 39A13; secondary 39B72

In this paper we have proposed a stochastic model for studying the dynamics of tuberculosis (TB) by incorporating vaccination of newly born babies. The total population in this model is subdivided in to four compartments, namely susceptible $S ( t )$ S ( t ) , infected $I ( t )$ I ( t ) , vaccinated newborns $V ( t ) $ V ( t ) , and recovered $R ( t ) $ R ( t ) . First, the developed model is expressed and analyzed by the deterministic approach. Since this approach neglects the randomness of the dynamics of the process, it has great limitations in the modeling process. To avoid this kind of issues, we transform the deterministic approach into a stochastic one, which is known to play a significant role by providing additional degree of realism compared to the deterministic approach. The analysis of the model is done employing both approaches. The invariant region, positivity of the solution, equilibrium points and their stability are checked. According to the analysis, we came to rea...

The present study deals with the analysis of a Lotka-Volterra model describing competition between tumor and immune cells. The model consists of differential equations with piecewise constant arguments and based on metamodel constructed by Stepanova. Using the method of reduction to discrete equations, it is obtained a system of difference equations from the system of differential equations. In order to get local and global stability conditions of the positive equilibrium point of the system, we use Schur-Cohn criterion and Lyapunov function that is constructed. Moreover, it is shown that periodic solutions occur as a consequence of Neimark-Sacker bifurcation.

In this paper it is shown that recursive local state estimators (related to Roesser models) can be constructed uaing global state observers. (related to column to column propagation models). These observers can be designed based on a result (pole assignment) of algebraic 2-D systems theory.

The purpose of the paper is to propose the Chebyshev spectral collocation method to solve a certain type of stochastic delay differential equations. Based on a spectral collocation method, the scheme is constructed by applying the differentiation matrix D N to approximate the differential operator d dt. D N is obtained by taking the derivative of the interpolation polynomial P N (t), which is interpolated by choosing the first kind of Chebyshev-Gauss-Lobatto points. Finally, numerical experiments are reported to show the accuracy and effectiveness of the method.

ln this paper, we study the boundedness, persistence, and the global asymptotic behavior of the positive solutions of the system of two difference equations Xn+l 1 + xr~ yr~ : --, Yn+l = 1 + --, n = O, 1,..., Yn-m Xn-m where xi, yi, i = -rn,-m + 1,... ,0 are positive numbers and m is a positive integer.

This paper introduces the rational forward difference operator for differential computation on a rational Bézier patch based on its control mesh. With this rational version of the forward difference operator, and by ignoring the appropriate dependent (on lower order derivatives) terms of various derivatives, the derivatives themselves have the same expressions as their polynomial counterparts; the curvature and torsion, and the first and second fundamental forms, all have very similar expressions as those equations from classical differential geometry. This new approach also allows straightforward generalization to higher dimensional rational Bézier patches, the special co-dimension 1 case of which is treated with the result of the first and second fundamental forms.

In this paper, by using the Lie symmetry analysis, all of the geometric vector fields of the ( 3 + 1 ) $(3+1)$ -Burgers system are obtained. We find the 1, 2, and 3-dimensional optimal system of the Burger system and then by applying the 3-dimensional optimal system reduce the order of the system. Also the nonclassical symmetries of the ( 3 + 1 ) $(3+1)$ -Burgers system will be found by employing nonclassical methods. Finally, the ansatz solutions of BS equations with the aid of the tanh method has been presented. The achieved solutions are investigated through two- and three-dimensional plots for different values of parameters. The analytical simulations are presented to ensure the efficiency of the considered technique. The behavior of the obtained results for multiple cases of symmetries is captured in the present framework. The outcomes of the present investigation show that the considered scheme is efficient and powerful to solve nonlinear differential equations that arise in t...

Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defined based on í µí»¼-cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first type was fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge set. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are discussed. Some interesting remarks on fuzzy chromatic polynomial of fuzzy graphs have been derived. Further, some results related to the concept are proved. Lastly, fuzzy chromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained.

A general theory of doubly periodic (DP) arrays over an arbitrary finite field GF(q) is presented. Fit the basic properties of DP arrays are examhed. Next modules of linear recurhg (LR) arrays are defined and their algebr$c properties discus& in connection with ideals in an extension ring R of t& ring R of bivariate polyuomials with coefficien@ iu GF(q). A finite R-module of DP arrays is shown to coinci@ with the R-module of LR arrays defined by a zero-dimensional ideal in R. Equivalence relations between DP arrays are explored, i.e., rearrangements of arrays by means of unimodular transformations. Decimation and interleaving of arrays are defined in a two-dimensional sense. The general theory is followed by application to irreducible LR arrays. Among irreducible arrays, M-arrays are a two-dimensional analog of M-sequences and may be constructed from M-sequences by means of unimodular transformations. The results of this paper are also important in studying properties of Abeliau codes.

The main purpose of this work is to provide an optimized version of the Newton divided-difference algorithm presented by [11] for the polynomial interpolation problem. This optimized algorithm is used for the computation of the determinant of a two-variable polynomial matrix.

An analytical delay model for BiCMOS driver circuits is presented. The model is based on physical device parameters and can be used to estimate both the pull-up and the pull-down times for a variety of circuit configurations. The intrinsic delay associated with the bipolar transistors is taken into consideration by using a charge control model that incorporates the high-injection effects

The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It has conjured up tremendous interest in mathematicians and non-mathematicians alike, who spent countless hours over millennia to explore its beauty and varied applications in science and engineering. The article attempts to record the pi exploration over centuries including its successive computation to ever increasing number of digits and its remarkable usages, the list of which is not yet closed.

Although recently there has been a great interest in studying of the behaviour of the solutions of rational difference equations, there are only a few papers devoted to systems of the rational difference equations. The aim of this study is to investigate the periodic character of the some systems of difference equations and to contribute the studies on this subject. This study consists of four sections.

In the first section; the definitions of difference equations are given.

In the second section; the studies on the systems of difference equations are summarized.
In the third section;  the periodic character of the some systems of difference equations are investigated.

In the fourth section; the systems of difference equations that investigated in third section are generalized.

Son zamanlarda rasyonel fark denklemlerinin çözümlerinin davranışları ile ilgili önemli çalışmalar yapılmasına rağmen , rasyonel fark denklem sistemleri ile ilgili az sayıda çalışma vardır. Bu çalışmanın amacı, bazı fark denklem sistemlerinin çözümlerinin periyodikliğini incelemek ve bu konudaki çalışmalara katkıda bulunmaktır. Bu amaçla hazırlanan çalışma dört bölümden oluşmuştur.

Birinci bölümde fark denklemleri ile ilgili tanımlar verilmiştir.

İkinci bölümde fark denklem sistemleri ile ilgili yapılan çalışmalar özetlenmiştir.

Üçüncü bölümde bazı fark denklem sistemlerinin çözümlerinin periyodikliği incelenmiştir.

Dördüncü bölümde, üçüncü bölümde incelenen fark denklem sistemleri genelleştirilmiştir.

Physiologically-based pharmacokinetic (PBPK) models have been used to describe the distribution and elimination characteristics of intravenous ethanol administration. Further, these models have been used to estimate the ethanol infusion profile required to prescribe a specific breath ethanol concentration time course in a specific human being, providing a platform upon which other pharmacokinetic and pharmacodynamic investigations are based. In these PBPK models, the equivalence of two different peripheral tissue models are shown and issues concerning the mass flow into the liver in comparison with ethanol metabolism in the liver are explained.

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

The aim of this paper is to study Jindalrae and Gaenari numbers and polynomials in connection with Jindalrae-Stirling numbers of the first and second kinds. For this purpose, we first introduce Jindalrae-Stirling numbers of the first and second kinds as extensions of the notions of the degenerate Stirling numbers of the first and second kinds, and deduce several relations involving those special numbers. Then we introduce Jindalrae and Gaenari numbers and polynomials and obtain some explicit expressions and identities associated with those numbers and polynomials. In addition, we interpret our results by using umbral calculus.

A stand growth model for Mediterranean maritime pine in the Iberian Peninsula (Pinus pinaster Ait.) is presented. The model consist of a site index submodel developed from 281 stem analysis and validated with the data from 92 permanent sample plots. Three height growth equations in difference form are tested and the Bailey and Clutter (1974) function is considered appropriate due to its good performance with both fitting and validation data. A compatible growth and yield submodel is then elaborated with the data from the Forest Research Centre (CIFOR-INIA) permanent sample plots network. The future state of the stand is determined by the current state, characterized by basal area, age and density. The model is completed with a control function that predicts the diameter after thinning and a mortality rate for non-thinned intervals. The global model was validated with data of 13 independent permanent thinning plots from two experimental sites. The simulation of the long-term projection of volume, basal area and DBH after thinning shows certain overestimation in diameter after thinning and in volume. However, the results show errors lower than 5% and little bias.

Recurrent Self-Organizing Map (RSOM) is studied in temporal sequence processing. RSOM includes a recurrent difference vector in each unit of the map, which allows storing temporal context from consecutive input vectors fed to the map. RSOM is a modification of the Temporal Kohonen Map (TKM). It is shown that RSOM learns a correct mapping from temporal sequences of a simple synthetic data, while TKM fails to learn this mapping. In addition, two case studies are presented, in which RSOM is applied to EEG based epileptic activity detection and to time series prediction with local models. Results suggest that RSOM can be efficiently used in temporal sequence processing.

This paper reveals a computational method based using a tau method with Jacobi polynomials for the solution of fuzzy linear fractional differential equations of order 0 < v < 1. A suitable representation of the fuzzy solution via Jacobi polynomials diminishes its numerical results to the solution of a system of algebraic equations. The main advantage of this method is its high robustness and accuracy gained by a small number of Jacobi functions. The efficiency and applicability of the proposed method are proved by several test examples.

Transmission dynamics of swine influenza pandemic is analysed through a deterministic model. Qualitative analysis of the model includes global asymptotic stability of disease-free and endemic equilibria under a certain condition based on the reproduction number. Sensitivity analysis to ponder the effect of model parameters on the reproduction number is performed and control strategies are designed. It is also verified that the obtained numerical results are in good agreement with the analytical ones.

A stand growth model for Mediterranean maritime pine in the Iberian Peninsula (Pinus pinaster Ait.) is presented. The model consist of a site index submodel developed from 281 stem analysis and validated with the data from 92 permanent sample plots. Three height growth equations in difference form are tested and the Bailey and Clutter (1974) function is considered appropriate due to its good performance with both fitting and validation data. A compatible growth and yield submodel is then elaborated with the data from the Forest Research Centre (CIFOR-INIA) permanent sample plots network. The future state of the stand is determined by the current state, characterized by basal area, age and density. The model is completed with a control function that predicts the diameter after thinning and a mortality rate for non-thinned intervals. The global model was validated with data of 13 independent permanent thinning plots from two experimental sites. The simulation of the long-term projection of volume, basal area and DBH after thinning shows certain overestimation in diameter after thinning and in volume. However, the results show errors lower than 5% and little bias.

Eddy current losses in conducting plates placed in a non-uniform alternating magnetic field at cryogenic temperatures have been calculated taking into account temperature dependencies of resistivity, heat capacity and heat-transfer coefficient. The calculation method is based on a coupled solution of a quasi-stationary electromagnetic problem and a nonstationary heat-conduction equation. A simplification of the task is achieved by neglecting eddy current reaction and free space charge. In this case the problem is solved in two stages: 1) the calculation of electromagnetic field; 2) the determination of a temperature distribution in the plate. As the first boundary condition for heat-conduction equation it is assumed that the heat transfer from edges of the plate equals zero. The second boundary condition is determined as a solution of the integral equation describing the law of conservation of energy relevant to the plate. The nonlinear two-dimensional heat conduction equation has been solved using partition of coordinates, implicit difference approximation and modified Thomas algorithm for obtained finite-difference equations. The numerical results remain in a good agreement with data obtained by means of an experimental model of the cryogenic device, while neglecting of thermal processes results in substantial errors amounting up to several hundreds percent.

The paper describes the fundamentals, analytical formulation and application of an efficient methodology for the transient and steady-state analysis of the synchronous machine. The state space phase co-ordinates model of the synchronous machine takes into account the timevarying nature of the machine inductances and inter-spatial harmonic inductances. The dynamics of the synchronous machine are represented by a set of ordinary differential equations (ODES) in the time domain. The machine behaviour is analysed under severe abnormal operation conditions. The model allows a detailed and accurate transient representation of the machine, which cannot he achieved with conventional models based on the dq0 frame of reference, especially for the case of unbalanced fault analysis. The steady state after the fault is cleared up is obtained efficiently with the application of a Newton technique for the acceleration of the convergence of the state variables to the limit cycle. The synchronous machine model is compared in terms of accuracy and computational efficicncy against a widely used and accepted machine model of an electro-magnetic transient program (EMTP-ATP).

We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a, k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C 0 -semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behavior of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley-Torvik equations. D α t u(t) = Au(t) + D α−1 t f (t, u(t)), t ∈ [0, T ], u(0) + g(u) = u 0 , u (0) = 0, Key words and phrases. Linear and semilinear evolution equations; regularized operator families; mild solutions.

The present study introduces a new version of homotopy perturbation method for the solution of system of fractional-order differential equations. In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the nonlinear problem. The systems include fractional-order stiff system, the fractional-order Genesio system, and the fractional-order matrix Riccati-type differential equation. The new approximate analytical procedure depends only on two components. Comparing the methodology with some known techniques shows that the present method is relatively easy, less computational, and highly accurate.

A new scheme is proposed that improves the solution at low energies, keeping the desired accuracy in the calculation of the mean quantities while saving a significant amount of CPU time. This is important in view of the applications of the method since the typical number of nodes to be used in the combined space-energy domain is in the range of 10 4 -10 5 . , in 1975, where he investigated the physical properties of the MOS structures and the problems of analytical modeling of semiconductor devices. Since 1978, he has been teaching an annual course of Quantum Electronics in the Faculty of Engineering of the same university. Since 1983 he has been working in a group involved in numerical analysis of semiconductor devices, acting as task leader in a number of EEC-supported Projects in the area of CAD for VLSI. In 1986, he was a Visiting Scientist, on a oneyear assignment, at the IBM Thomas J. Watson Research Center, Yorktown Heights, NY, studying the discretization techniques for the higher order moments of the Boltzmann Transport Equation. He was with IBM for shorter periods afterwards, investigating and implementing impact-ionization models based on the average energy of the carriers. In 1990, he was appointed Full Professor of Microelectronics at the University of Bologna.

Wind turbines may have an important impact on power quality. Flicker is a more serious issue for fixed-speed wind turbines, because these turbines produce electric power following the variations of the incident wind. During continuous operation, wind variations will result in power fluctuations and consequently in voltage fluctuations. It is necessary to evaluate wind turbine flicker emission level, and traditionally time domain simulations have been used to perform the analysis. This paper presents a complete frequency domain model to study flicker produced during wind turbine continuous operation. The model includes a realistic wind speed model as observed by the wind turbine, and also a frequency domain induction generator model is presented. The frequency domain model is compared with a time domain model. The frequency domain approach, as shown in the paper, may be very useful for flicker analysis in electric networks.

ÖZ Bu çalışmada Celal Bayar Üniversitesi'nde 2014-2015 eğitim-öğretim döneminde formasyon eğitimine katılan öğretmen adaylarının Bilgisayar Destekli Öğretime ilişkin tutumları ile demografik bazı değişkenler arasındaki ilişkiler incelenmiştir. Bu amaçla, 478 öğretmen adayına Arslan (2006) tarafından geliştirilen Bilgisayar Destekli Eğitim Yapmaya İlişkin Tutum Ölçeği uygulanmıştır. Sonuç olarak, formasyon eğitimine katılan öğretmen adaylarının bilgisayar destekli öğretime yönelik tutumlarının olumlu yönde olduğu, adayların bilgisayar destekli öğretim yapmaya yönelik tutumlarında cinsiyete veya öğretmenlik deneyimlerine bağlı anlamlı farklılık olmadığı; ancak, yaşa bağlı olarak bilgisayar destekli öğretim yapmaya ilişkin tutumlar incelendiğinde 30 yaş üstü öğretmen adaylarının 25 yaş altı adaylara göre pozitif yönde anlamlı derecede olumlu tutumda oldukları görülmüştür. Ayrıca, İlahiyat alanından mezun öğretmen adaylarının tutumlarında Fizik, Kimya, Matematik ve Türk Dili ve Edebiyatı alanlarından mezun adaylara göre pozitif yönde anlamlı farklılık gözlenmiştir. Son olarak Bilgisayar Destekli Öğretim/Eğitim derslerinin uygulamalı olarak yapılması gerektiği önerilmiştir. Anahtar Kelimeler: BDE tutum ölçeği, formasyon öğretmen adayları, bilgisayar destekli öğretim.

Integral transform methods are widely used to solve the several dynamic equations with initial values or boundary conditions which are represented by integral equations. With this purpose, the Sumudu transform is introduced in this article as a new integral transform on a time scale &quot;Equation missing&quot; to solve a system of dynamic equations. The Sumudu transform on time scale &quot;Equation missing&quot; has not been presented before. The results in this article not only can be applied on ordinary differential equations when T = ℝ , difference equations when T = ℕ 0 , but also, can be applied for q-difference equations when T = q ℕ 0 , where q ℕ 0 : = { q t : t ∈ ℕ 0  for  q &gt; 1 } or T = q ℤ ¯ : = q ℤ ∪ { 0 } for q &gt; 1 (which has important applications in quantum theory) and on different types of time scales like T = h ℕ 0 , T = ℕ 0 2 and T = T n the space of the harmonic numbers. Finally, we give some applications to illustrate our main results. 2010 Mathematics Subj...

This work brings together classical polynomial theory as it relates to the Levinson recurrence for a Hermitian Toeplitz operator and matrix theory as it relates to the class of Hermitian centro-Hermitian matrices. A new computationally efficient alternative is presented to the Levinson recurrence on either the Hermitian or skew-Hermitian polynomial spaces. This approach also leads to an entirely new algorithm for solving systems of linear equations when the coefficient matrix is Hermitian Toeplitz or real symmetric Toeplitz. Analysis of the computational complexity of the algorithms presented is also performed, and it is shown that these algorithms lead to significant improvements in the computational complexity as compared to the previously best-known recursive algorithms. They also provide further insight into the mathematical properties of the structurally rich Toeplitz matrices.

A study of the finite difSerence solution of the nonlinear partial differential equations governing two-and three-dimensional semiconductor devices is conducted on a SIMD computer. This nonlinear system is solved using Jacobi iteration and successive-under-relaxation. Row scaling and a zero order regularizer are used to aid in convergence. On a 16K CM-2 problems with up to 16.7 million unknowns have been solved. Problems of this size have not previously been reported. The ability to accurately model larger and more realistic three-dimensional devices is necessary to gain a greater physical understanding of their behavior.

This paper focuses on the existence, uniqueness and global exponential stability of periodic solution for Cohen-Grossberg neural networks (CGNN) with periodic coefficients and time-varying delays. Some novel delayindependent criteria are obtained by using contraction mapping theorem and comparison principle. The present results improve and extend those in many publications, and shows that under some delay-independent criteria, the CGNNs may admit a periodic solution, which is globally exponential stable via impulsive controller even if it is originally unstable or divergent. Two examples with numerical simulations are given to demonstrate the efficiency of theoretical results.

In this paper, we consider the (3 + 1)-dimensional time-fractional Schamel-Zakharov-Kuznetsov-Burgers (SZKB) equation. With the help of the Riemann-Liouville derivatives, the Lie point symmetries of the (3 + 1)-dimensional time-fractional SZKB equation are derived. By applying the Lie point symmetry method as well as Erdélyi-Kober fractional operator, we get the similarity reductions of the time-fractional SZKB equation. Conservation laws of the time-fractional SZKB are constructed. Moreover, we obtain its power series solutions with the convergence analysis. In addition, the analytical solution is obtained by modified trial equation method. Finally, stability is analyzed graphically in different planes.