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.

This paper explores the approximate analytical solution of nonlinear Klein-Gordon equations (NKGE) by using Multistep Modified Reduced Differential Transform Method (MMRDTM). Through this proposed strategy, the nonlinear term is substituted by associating Adomian polynomials obtained by utilization of a multistep approach. The NKGE solutions can be obtained with a reduced number of computed terms. In addition, the approximate solutions converge rapidly in a wide time region. Three examples are provided to illustrate the effectiveness of the proposed method to obtain solutions for the NKGE. Graphical results are shown to represent the behavior of the solution so as to demonstrate the validity and accuracy of the MMRDTM.

In this paper, we propose and execute the Multistep Modified Reduced Differential Transform Method (MMRDTM) to obtain the solution of fractional nonlinear Schrodinger equations (FNLSEs). Using this proposed technique, the nonlinear term is substituted by the related Adomian polynomials followed by application of a multistep approach. In addition, the fractional NLSE solutions can be obtained with less computational effort than other approximate analytical solution such as Adomian decomposition method, homotopy pertubation method, variational iteration method and differential transformation method. Besides that, it offers precise estimated solutions over a longer time period. We considered a few FNLSEs and represent the features of these solutions in the form of graphs to demonstrate the power and precision of the MMRDTM.

In this study, we considered time-fractional nonlinear telegraph equations (TFNLTEs). For solving the TFNLTEs, we deployed a method known as the Multistep Modified Reduced Differential Transform Method (MMRDTM). Prior to the multistep technique, the nonlinear term in TFNLTEs is replaced with corresponding Adomian polynomials. It can be observed that the MMRDTM is much simpler and more straightforward. On top of that, it works exceptionally where the obtained solutions are more accurately approximated over time. To demonstrate the performance of the MMRDTM in terms of its capabilities and accuracies, we provided two numerical examples of solving TFNLTEs by using MMRDTM and Modified Reduced Differential Transform Method (MRDTM). By comparing the absolute errors of the obtained solutions by both methods, we demonstrated that the solutions provided by the MMRDTM much closer to the exact solutions compared to the corresponding solutions yielded by the MRDTM. This justified that the MMRDTM provides highly accurate and precise solutions for the TFNLTEs.

We may secure the content of text, audio, image and video during their transmission from one party to another party via an open channel such as the internet by using cryptograph. Logistic-Sine System (LSS) is a combination on two 1D chaotic maps which are Logistic Map and Sine Map. By applying the LSS into cryptography, the image encryption and decryption can be performed. This study is focusing on the performance test of the image encryption and decryption processes by using the LSS. For comparison purpose, we compare the performance of the encryption and decryption by using two different chaotic systems, which are the LSS and Logistic-Tent System (LTS). The result shows that system with LSS is less efficient than LTS in term of encryption time but both systems have similar efficiency in term of decryption time.

The study of solitons and compactons is important in nonlinear physics. In this paper we combined the Adomian polynomials with the multi-step approach to present a new technique called Multi-step Modified Reduced Differential Transform Method (MMRDTM). The proposed technique has the advantage of producing an analytical approximation in a fast converging sequence with a reduced number of calculated terms. The MMRDTM is presented with some modification of the Reduced Differential Transformation Method (RDTM) with multi-step approach and its nonlinear term is replaced by the Adomian polynomials. Therefore, the nonlinear initial value problem can easily be solved with less computational effort. Besides that, the multi-step approach produces a solution in fast converging series that converges the solution in a wide time area. Two examples are provided to demonstrate the capability and benefits of the proposed method for approximating the solution of NKdVEs with compactons. Graphical inputs are used to represent the solution and to demonstrate the precision and validity of the MMRDTM in graphic illustration. From the results, it was found that it is possible to obtain highly accurate results or exact solutions by using the MMRDTM.