On the solution of fractional order nonlinear boundary value problems by using differential transformation method.

In this research, we study about fractional order for nonlinear of fifth-order boundary value problems and produce a theorem for higher order of fractional of nth-order boundary value problems. The aim of this study was to evaluate and validate the theorem and provide several numerical examples to test the performance of our theorem. We also make comparison between exact solutions and differential transformation method(DTM) by calculating the error between them. It is shown that DTM has very small error and suitable in several numerical solutions since it is effective and provide high accuracy

On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and adomian decomposition method.

We study higher-order boundary value problems (HOBVP) for higher-order nonlinear differential equation. We make comparison among differential transformation method (DTM), Adomian decomposition method (ADM), and exact solutions. We provide several examples in order to compare our results. We extend and prove a theorem for nonlinear differential equations by using the DTM. The numerical examples show that the DTM is a good method compared to the ADM since it is effective, uses less time in computation, easy to implement and achieve high accuracy. In addition, DTM has many advantages compared to ADM since the calculation of Adomian polynomial is tedious. From the numerical results, DTM is suitable to apply for nonlinear problems

Solving eighth-order boundary value problems using differential transformation method

In this study, we solved linear and nonlinear eighth-order boundary value problems using Differential Transformation Method. Then we calculate the error of DTM and compare the results with other methods such as modified application of the variational iteration method (MVAM), homotopy perturbation method (HPM) and modified Adomian decomposition method (MADM). We compared the errors of each method with exact solutions. We provided several numerical examples in order to show the accuracy and efficiency of present method. The results showed that the DTM is more accurate in comparison with those obtained by other methods

Comparing linear and nonlinear differential equations of differential transformation method by other numerical methods

In this study, we solve fifth-order boundary value problems by using the DTM for linear and nonlinear differential equations and compare the results with other methods such as Adomian Decomposition Method (ADM), Noor Decomposition Method and Variational Iteration Method. We provide several numerical examples in order to show the accuracy of the method. Further, we also solve sixth-order nonlinear boundary value problems and compare the result to ADM. The present study shows that the DTM is able to provide good results with high accuracy and the method is also easy to apply

Analytical solutions of nonlinear Schrodinger equations using multistep modified reduced differential transform method

This paper aims to propose and implement the Multistep Modified Reduced Differential Transform Method (MMRDTM) to find solution of Nonlinear Schrodinger Equations (NLSEs). Through the proposed technique, we replaced the nonlinear term in the NLSEs by the corresponding Adomian polynomials prior applying the multistep approach. Thus, we can obtain solutions for the NLSEs in easier way with less complexity. In addition, the solutions can be approximated more accurately over a longer time frame. We considered several NLSEs and illustrate the features of these solutions in the form of graphs in order to show the power and accuracy of the MMRDTM

Elgamal digital signature scheme with integrated CFEA-technique

One of the four security goals is authentication. Authentication is a mechanism to ensure that we are communicating with the intended party. If Alice and Bob want to communicate securely, then the authentication mechanism will be able to ensure that Alice is truly communicating with Bob, and Bob is truly communicating with Alice. This mechanism can be provided by the cryptography. One of the most established cryptography schemes is ElGamal cryptosystem. The original version of this cryptosystem is to provide confidentiality through encryption and decryption procedures. By manipulating these procedures, the authentication mechanism can be carried out. Thus, ElGamal Digital Signature Scheme emerges as one of the most popular authentication mechanisms. In order to provide good level of security, proper parameters must be used in this scheme. This includes the size of the parameters. Larger parameters will provide a better level of security. As a consequence, the performance of the scheme becomes an issue in real life application. In this paper, we proposed the enhancement of the ElGamal Digital Signature Scheme by integrating the Continued-Fraction-Euclidean-Algorithm (CFEA) technique. This technique is able to reduce the number of data to be processed in the signing and verification procedures. By integrating the CFEA-technique into the ElGamal Digital Signature Scheme, any number of documents can be compressed becomes a pair of documents. Therefore, the signing and verification procedures can be done in smaller number of steps

Differential transformation method for solving sixth-order boundary value problems of ordinary differential equations

In this study, sixth-order boundary value problems for linear and nonlinear differential equations have been solved by using Differential Transformation Method (DTM). The numerical solutions are given in several examples. For each example, the solution given by DTM is compared with the exact solution. Absolute relative error (ARE) for each iteration can be computed. Therefore, the maximum absolute relative error (MARE) of the DTM can be obtained. To show that the solution given by the DTM has higher level of accuracy, the absolute relative error of the DTM has been compared with the other methods such as Adomian decomposition method with Green’s function, modified decomposition method (MDM), homotopy perturbation method (HPM), Variational Iteration Method (VIM) and Quintic B-Spline Collocation Method. Comparison graphs are given at the end of this paper. The obtained result shows that the proposed method is able to provide better approximation in term of accuracy

Elgamal Digital Signature Scheme With Integrated CFEA-Technique

One of the four security goals is authentication. Authentication is a mechanism to ensure that we are communicating with the intended party. If Alice and Bob want to communicate securely, then the authentication mechanism will be able to ensure that Alice is truly communicating with Bob, and Bob is truly communicating with Alice. This mechanism can be provided by the cryptography. One of the most established cryptography schemes is ElGamal cryptosystem. The original version of this cryptosystem is to provide confidentiality through encryption and decryption procedures. By manipulating these procedures, the authentication mechanism can be carried out. Thus, ElGamal Digital Signature Scheme emerges as one of the most popular authentication mechanisms. In order to provide good level of security, proper parameters must be used in this scheme. This includes the size of the parameters. Larger parameters will provide a better level of security. As a consequence, the performance of the scheme becomes an issue in real life application. In this paper, we proposed the enhancement of the ElGamal Digital Signature Scheme by integrating the Continued-Fraction-Euclidean-Algorithm (CFEA) technique. This technique is able to reduce the number of data to be processed in the signing and verification procedures. By integrating the CFEA-technique into the ElGamal Digital Signature Scheme, any number of documents can be compressed becomes a pair of documents. Therefore, the signing and verification procedures can be done in smaller number of steps

Reduction-by-percentage compression technique for reducing sizes of plaintext prior to encryption algorithm

Other than security, another major concern in cryptography is efficiency to ensure cryptosystem could be embedded and deployed in various communication devices. Encrypting high numbers of data would consume high computational and storage capacity costs. These issues could affect the efficiency of cryptosystem. One of the approaches to overcome these issues is by integrating data compression technique into cryptosystem. To avoid any encryption and decryption error, lossless compression techniques are deployed in cryptography. The compression techniques are deployed to reduce either the size or number of plaintexts prior to encryption algorithm. Nevertheless, the sizes of the compressed plaintext are still large. To deal with this issue, we proposed a simple technique with ability for reducing the sizes of the compressed plaintext. The inverse of this technique is able to recover the original value of the data without any loss or difference compared to the original data. With smaller sizes, the encryption algorithm would process inputs with smaller sizes, and these could potentially make the encryption algorithm be executed in cheaper computational and storage capacity costs

Approximate analytical solutions of nonlinear hyperbolic partial differential equation

The Multistep Modified Reduced Differential Transform Method (MMRDTM) is proposed and implemented in this study to obtain solutions of hyperbolic partial differential equations. We examine at the nonlinear Schrodinger equation (NLSE). Prior to implementing the multistep strategy, we switched the nonlinear term in the NLSE with the corresponding Adomian polynomials using the proposed technique. As a result, we can acquire solutions for the NLSE in a simpler and less difficult manner. Furthermore, the solutions can be estimated more precisely over a longer time period. We studied the NLS equation and graphed the features of this solution to show the strength and accurateness of the proposed technique