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In this work, we examine a fractal vehicular traffic flow problem. The partial differential equations describing a fractal vehicular traffic flow are solved with the aid of the local fractional homotopy perturbation Sumudu transform scheme and the local fractional reduced differential transform method. Some illustrative examples are taken to describe the success of the suggested techniques. The results derived with the aid of the suggested schemes reveal that the present schemes are very efficient for obtaining the non-differentiable solution to fractal vehicular traffic flow problem.

We integrate software testing within a broader context of specification, verification and testing.We present software testing as a structured, multi-phased activity.We precise milestones and deliverables.We present testing as a goal-oriented activity, with specific verifiable claims.Specific verifiable claims are based on engineering methods and techniques. Ever since it emerged in the late (nineteen) sixties, the discipline of software engineering has set itself apart from other engineering disciplines in a number of ways, including: the pervasiveness of its products; the complexity of its products and processes; the criticality of its applications; the difficulty of managing its processes and estimating its costs; the volatility of its workforce; the intractability of its process lifecycles; etc. A number of principles have emerged from recent software engineering research, that have the potential to bring a measure of control to the practice of this discipline; but they have not made it into routine practice in industry. We argue that the classroom is a good place to start acquainting students with these principles, and to start getting them into the habit of adhering to them as a matter of routine practice.

We presented a unified description of flow control and single steps of a program is given to obtain flexible definitions of algebraic manipulations. This is achieved by using the notion of relational diagram. We show how the notion of relational diagram, introduced by Schmidt, can be used to give a demonic definition for a wide range of programming constructs. It is shown that the input-output relation of a compound diagram is equal to that of the diagram in which each sub-diagram has been replaced by its input-output relation. This process is repeated until elementary diagrams is obtained.

In this paper, we analyze the dynamic behavior of the ill-posed Boussinesq equation (IPBE) that arises in nonlinear lattices and also in shallow water waves. Some solitary wave solutions are obtained by using the solitary wave ansatz method and the Bernoulli sub-Ode. By applying the technique of nonlinear self-adjoint, a quasi self-adjoint substitution for the IPBE is constructed. The classical symmetries of the equation are constructed. Then, we used along with the obtained nonlinear self-adjoint substitution to construct a set of new conservation laws (Cls).

We consider a class of nonlinear initial value problems with delay. Using an abstract fixed point theorem, we prove an existence result producing a unique bounded solution.

We prove the existence of a g-best proximity point for a pair of mappings, by using suitable hypotheses on a metric space. Moreover, we establish some convergence results for a variational inequality problem, by using the variational characterization of metric projections in a real Hilbert space. Our results are applicable to classical problems of optimization theory.

The computer-aided diagnosis has become one of the major research topics in medical diagnostics. In this research paper, we focus on designing an automated computer diagnosis by combining two major methodologies, namely the fuzzy base systems and the evolutionary genetic algorithms and applying them to the Saudi Arabian breast cancer diagnosis database, to be employed for assisting physicians in the early detection of breast cancers, and hence obtaining an early-computerized diagnosis complementary to that by physicians. Our hybrid algorithm, the genetic-fuzzy algorithm, has produced optimized diagnosis systems that attain high classification performance, in fact, our best three rule system obtained a 97% accuracy, with simple and well interpretive rules, and with a good degree of confidence of 91%.

In this chapter, the authors categorize methods that are used to formally specify and verify software requirements. They discuss several formal method-related subjects such as calculus fuzzy and relational calculus.

We consider that nondeterministic programs behave as badly as they can and loop forever whenever they have the possibility to do so. We deal with a relational algebra model to define a nondeterministic refinement fuzzy ordering (nondeterministic fuzzy inclusion) and also the associated fuzzy operations which are fuzzy nondeterministic join (\(\sqcup_{fuz}\)), fuzzy nondeterministic meet (\(\sqcap_{fuz}\)), and fuzzy nondeterministic composition (\({\square \mbox{\tiny}}_{fuz}\)). We also give some properties of these operations and illustrate them with simple examples.

In relational semantics, the input-output semantics of a program is a relation on its set of states. We generalise this in considering elements of Kleene algebras as semantical values. In a nondeterministic context, the demonic semantics is calculated by considering the worst behaviour of the program. In this paper, we concentrate on while loops. Calculating the semantics of a loop is difficult, but showing the correctness of any candidate abstraction is much easier. For deterministic programs, Mills has described a checking method known as the while statement verification rule. A corresponding programming theorem for nondeterministic iterative constructs is proposed, proved and applied to an example. This theorem can be considered as a generalisation of the while statement verification rule to nondeterministic loops. The paper generalises earlier relation-algebraic work to the setting of modal Kleene algebra, an extension of Kozen’s Kleene algebra with tests that allows the internalisation of weakest liberal precondition and strongest liberal postcondition operators.

The non-differentiable solution of the linear and non-linear partial differential equations on Cantor sets is implemented in this article. The reduced differential transform method is considered in the local fractional operator sense. The four illustrative examples are given to show the efficiency and accuracy features of the presented technique to solve local fractional partial differential equations.

We deal with a relational model for the demonic semantics of programs. The demonic semantics of a while loop is given as a fixed point of a function involving the demonic operators. This motivates us to investigate the fixed points of these functions. We give the expression of the greatest fixed point with respect to the demonic ordering ({\it demonic inclusion}) of the semantic function. We prove that this greatest fixed coincides with the least fixed point with respect to the usual ordering ({\it angelic inclusion}) of the same function. This is followed by an example of application.

We present a generalization of a theorem of Mills, known as the while statement verification rule, that can be used to verify that a deterministic while loop computes a given function f. First, we study an abstract form of (nondeterministic) looping in the setting of relation algebras. We state a theorem showing how to verify that a given relation is the relation computed by the abstract loop. Then, we specialize this theorem to three forms of generalized looping structures that have been proposed in the literature, namely Dijsktra's do od, Anson's do upon and Parnas' it ti. We take a demonic viewpoint; that is, we assume the worse possible execution of the nondeterministic programs under consideration.

We present a refinement ordering between binary relations, viewed as programs or specifications. This ordering induces a complete join semilattice that can be embedded in a relation algebra. This embedding then allows an easy proof of many properties of the refinement semilattice, by making use of the well-known corresponding properties of relation algebras. The operations of the refinement semilattice corresponding to join and composition in the embedding algebra are, respectively, demonic join and demonic composition. The weakest prespecification and postspecification operators of Hoare and He, defined over a relation algebra, also have corresponding operators in the semilattice.

We study a family of random differential equations with boundary conditions. Using a random fixed point theorem, we prove an existence theorem that yields a unique random solution.